Talk:Eigenvalues and eigenvectors/Archive 2

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Movements

Movements are affine transformations, this is true. Affine transformations are a more general class, in

which the new reference frame is, in general, not necessarily the same as the initial reference frame. The necessary and sufficient conditions for a transformation to be linear or not are additivity and homogeneity. Geometrically, planar movements are classified as three types: those that preserve the direction of infinitely many lines (translations), those that preserve the direction of exactly one line (shears: horizontal and vertical), and those that preserve only one point (rotations). Of those, shear and rotation are linear transformations in non-homogenous coordinates. Translation is linear transformation in homogenous coordinates. In addition to those three, affine transformations include scaling, which is not movement.

Do you agree with the above? If not, what is wrong? --Lantonov (talk) 08:22, 19 May 2008 (UTC)

Translation is linear transformation in homogenous coordinates.

Again, you're writing sentences that don't mean anything. In vector space theory the concept of "homogenous coordinates" has absolutely no relevance. Sentences don't become true just because you add a lot more irrelevant jargon. Vector spaces are simple structures. Their properties do not depend on your choice of representation.

Translations are not linear maps, because they are not additive (except for translation by zero). Please don't discuss them in this article; they simply do not belong here.

You are simply not allowed to move the origin of vector spaces about, only change the bases with respect to the same origin. The zero element of a vector space is not dependent on the choice of basis. Bases are not the same as co-ordinate systems.

End of discussion.

Please stop writing articles about subjects you simply don't understand. -- GWO (talk)

This was helpful. I will not answer immediately, need specialist in projective geometry and projective spaces for this. For starters, check this: "Since a translation is an affine transformation but not a linear transformation, homogeneous coordinates are normally used to represent the translation operator by a matrix and thus to make it linear." in Translation (geometry). Mistake? --Lantonov (talk) 13:53, 21 May 2008 (UTC)

It's wrong in this context, yes. You can write a translation like that, as a matrix, but when you do it ceases to be a map on a vector space. Vector spaces are closed under addition and multiplication so the set of all vectors $(w_{1},w_{2},w_{3},1)$ does not form a vector space since (a) 0 is not an element and (b) it's not closed under addition, (c) it's not closed under scalar multiplication. We gain that trick to represent translations as matrices, but we lose the ability to add vectors in any meaningful sense. Since that representation breaks vector addition, we also break the definition of linear. You agree that linearity means

$T(x+y)=T(x)+T(y)$ and

$T(\lambda x)=\lambda T(x)$?

That's the very definition of linearity, right?

Let T be a translation by p:

$T(u+v)=u+v+p$ right? But

$T(u)+T(v)=(u+p)+(v+p)=u+v+2*p$.

$T(\lambda x)=\lambda x+p$ right? But

$\lambda T(x)=\lambda (x+p)=\lambda x+\lambda p.$.

Forget about matrix representations, are you really telling me that's a linear map?

It violates both the axioms that define linearity.

If that's a linear map, I'm Ann Coulter. -- GWO (talk)

The above is truer than true. Translation defies both conditions of linearity: additivity and homogeneity. I will not start arguing against that. However, there are some things that are nagging. In the matrix representation of translation in the plane (a 3x3 matrix, I don't have time to write it here), the eigenvalue is 1 with algebraic multiplicity 3. It generates 2 non-zero eigenvectors that must be interpreted in some way, probably in terms of lines at infinity (2-dimensional objects in projective geometry). I am not going to do it because projective geometry is inherently distasteful to me. I may heed your advice and go on to more interesting subjects. --Lantonov (talk) 06:10, 22 May 2008 (UTC)

Somehow, I got interested in this subject (a fleeting interest though), and dug out some more. In projective geometry, we have a class of elements at infinity: point at infinity (one-dimensional), line at infinity (two-dimensional), and one plane at infinity (three-dimensional) which is the whole space. Each element at infinity is represented by infinitely many finite elements. Finite elements are the usual elements of Euclidean geometry -- points, lines, and planes. So, point at infinity is represented by infinitely many parallel lines, and line at infinity is represented by infinitely many parallel planes. All finite parallel lines intersect at a point at infin ity, and all finite parallel planes intersect at a line at infinity. Finite elements that are representatives of one element at infinity are exactly one and the same element -- the element at infinity. Thus, all parallel lines are a point at infinity, and all parallel planes are a line at infinity. In homogenous coordinates, this central concept is expressed like this:

In the plane, all elements are written with 3 coordinates: (x/r, y/r, r) where x and y are the usual coordinates and / is division. r is any real number. Because we have division, there are 2 important cases: 1) r is 0 and 2) r is not 0. If r is 0, the element is at infinity. If r is not 0, the element is finite. There are infinitely many elements for which r is not 0, but only one element for which r is 0. Those infinitely many finite elements with r not 0 are representatives of that one element with r = 0. In coordinates: (x, y, 1) is exactly the same as, e.g. (x/2, y/2, 2). This is all that is needed initially to treat translation.

Going to vectors: ${\begin{pmatrix}x\\y\\1\end{pmatrix}}$ and ${\begin{pmatrix}2x/2\\2y/2\\2\end{pmatrix}}$ is one and the same vector: a representative of a vector that lies in a point at infinity (which is a one-dimensional object): the vector ${\begin{pmatrix}x\\y\\0\end{pmatrix}}$. Check the conditions for additivity and homogeneity for translation with such vectors (the finite ones: r not 0) and you will see that both conditions are satisfied if you have in mind the above. I knew this before but I have difficulty in interpretation of the 2 eigenvectors of translation: ${\begin{pmatrix}1\\0\\0\end{pmatrix}}$ and ${\begin{pmatrix}0\\1\\0\end{pmatrix}}$. Anyway, I do not intend to pursue this topic. --Lantonov (talk) 09:18, 22 May 2008 (UTC)

You're overcomplicating everything again. It's really nothing to do with projective geometry (another subject of which you clearly have only a sufficiently-superficial understanding to write long rambling incoherent paragraphs. A little knowledge is a dangerous things, especially among over-eager wikipedians.).

The problem is that the switch to and from homogeneous coordinates is not itself a linear map (because of the "1" in the last co-ordinate). So when you compose your linear matrix operations with this non-linear map, the overall transformation is not linear.

I've no interest in discussing projective geometry with someone who doesn't know what a vector space is, or what linear maps are, so go and annoy the projective geometry people. Or, better, go work on some biology articles, as your CV suggests you do actually know something about biology. -- GWO (talk)

"A linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The term "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms)." The linear map (since you prefer this term over transformation or operator) ${\begin{pmatrix}1&0&a\\0&1&b\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x+a\\y+b\\1\end{pmatrix}}$ transforms the vector ${\begin{pmatrix}x\\y\\1\end{pmatrix}}$ into vector ${\begin{pmatrix}x+a\\y+b\\1\end{pmatrix}}$. Both vectors are in the same vector space, so this linear map is endomorphism. Now tell me, where do you see here "a switch to and from homogeneous coordinates"? Also, how can I "compose a linear matrix operations with a non-linear map" when "linear matrix operation" (more exactly, operator) is the same as "linear map". And this time try to be more polite because my patience with you is running out. --Lantonov (talk) 12:08, 22 May 2008 (UTC)

About vector space:

1) associative

${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+({\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}+{\begin{pmatrix}w_{1}\\w_{2}\\1\end{pmatrix}})=({\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}})+{\begin{pmatrix}w_{1}\\w_{2}\\1\end{pmatrix}}$

2) commutative

${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}={\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}+{\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}$

3) identity element (there is only one zero vector over the whole vector space)

${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}0\\0\\0\end{pmatrix}}={\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}$

4) inverse elements

${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}-u_{1}\\-u_{2}\\-1\end{pmatrix}}={\begin{pmatrix}0\\0\\0\end{pmatrix}}$

5) distributivity for scalar multiplication over vector addition:

$a({\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}})=a{\begin{pmatrix}u_{1}\\v_{2}\\1\end{pmatrix}}+a{\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}$

6) distributivity for scalar multiplication over field (better: ring) addition:

$(a+b){\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}=a{\begin{pmatrix}u_{1}\\v_{2}\\1\end{pmatrix}}+b{\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}$

7) scalar multiplication is compatible with multiplication in the field (better: ring) of scalars:

$a(b{\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}})=(ab){\begin{pmatrix}u_{1}\\v_{2}\\1\end{pmatrix}}$

8) identity element:

$1{\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}={\begin{pmatrix}u_{1}\\v_{2}\\1\end{pmatrix}}$

Axioms of closure:

9) ${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}\in V$

10) $a{\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}\in V$

I do not need "a switch to and from homogeneous coordinates" in defining the vector space either. The definition in vector space is incorrect in one place though: "vector space over the field F". We do not need a field of scalars (no need for a⁻¹) to define a vector space, it is sufficient to have a ring with multiplicative unit. In this respect, the definition in Korn & Korn, section 12.4-1 is better. You can see from this how that statement of yours: Vector spaces are closed under addition and multiplication so the set of all vectors $(w_{1},w_{2},w_{3},1)$ does not form a vector space since (a) 0 is not an element and (b) it's not closed under addition, (c) it's not closed under scalar multiplication is plain wrong. --Lantonov (talk) 15:30, 22 May 2008 (UTC)

What is it that you claim to be a vector space V here?

Is it the set of vectors

${\begin{pmatrix}x\\y\\z\end{pmatrix}}$

or the set

${\begin{pmatrix}x\\y\\1\end{pmatrix}}$?

Because if its the former, then surely

${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+({\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}={\begin{pmatrix}u_{1}+v_{2}\\u_{2}+v_{2}\\2\end{pmatrix}}$?

If it's the former, you've not got closure, unless you now claim that $1+1=1$, which might (and I stress, might) be your most wrong-headed suggestion yet.

If it's the latter then your zero element is not in the space.

Either way, you've not defined a vector space.

If you define precisely your set, how to add two such vectors and how to multiply a vector and a scalar. I'll be able to tell you more precisely why you're wrong. At the moment, you're molding your definitions on the fly.

Oh, and you absolutely need a field for a vector space. Otherwise, you've just got a module (mathematics). And modules are a pain, because you can't find bases very easily -- GWO (talk)

The third element is NOT the coordinate z, it is any real number.

This doesn't mean anything. I'm not hung up on $z$. We can call it $q$ if you like.

Is your space the set of vectors

${\begin{pmatrix}x\\y\\q\end{pmatrix}}$ where $x,y,q$ are real numbers. If so, what is your concept of vector addition in this space?

All vectors in the definition for vector space are vectors in the plane, and they have only two non-homogenous coordinates. So this is the set ${\begin{pmatrix}x\\y\\1\end{pmatrix}}$ and not the set ${\begin{pmatrix}x\\y\\z\end{pmatrix}}$. And the zero element is in the space because vectors of the type ${\begin{pmatrix}x\\y\\0\end{pmatrix}}$ are in the space. Adding of vectors and multiplication by scalars is the same as in non-homogenous coordinates. "Module over a ring" is a generalization of vector space which is more general than module over a ring with multiplicative identity. --Lantonov (talk) 16:48,

`22 May 2008 (UTC)

So, all the vectors in the set are of the form ${\begin{pmatrix}x\\y\\1\end{pmatrix}}$ ... but ${\begin{pmatrix}x\\y\\0\end{pmatrix}}$ is in the set? What? Can you not see that that's completely contradictory? (And, again, you haven't defined vector addition or scalar multiplication).

You can, of course define a vector space by the set

${\begin{pmatrix}x\\y\\1\end{pmatrix}}$ with addition defined by

${\begin{pmatrix}x\\y\\1\end{pmatrix}}+{\begin{pmatrix}u\\v\\1\end{pmatrix}}={\begin{pmatrix}x+u\\y+v\\1\end{pmatrix}}$ and scalar multiplication

$\lambda {\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}\lambda x\\\lambda y\\1\end{pmatrix}}$

That's a perfectly well defined vector space (its just a copy the real plane, with a completely superfluous "1" attached to every pair of co-ordinates). Your zero element is just

${\begin{pmatrix}0\\0\\1\end{pmatrix}}$

Only get this: because you're not using the standard definition of "addition" on the third component, multiplication by a 3x3 matrix does not define a linear map. Matrix multiplication is only linear when we use the standard rules of vector addition!

Check this:

${\begin{pmatrix}1&0&a\\0&1&b\\0&0&1\end{pmatrix}}{\begin{pmatrix}x+u\\y+v\\1\end{pmatrix}}={\begin{pmatrix}x+u+a\\y+v+b\\1\end{pmatrix}}$ but

${\begin{pmatrix}1&0&a\\0&1&b\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}+{\begin{pmatrix}1&0&a\\0&1&b\\0&0&1\end{pmatrix}}{\begin{pmatrix}u\\v\\1\end{pmatrix}}={\begin{pmatrix}x+a\\y+b\\1\end{pmatrix}}+{\begin{pmatrix}x+a\\y+b\\1\end{pmatrix}}={\begin{pmatrix}x+u+2a\\y+v+2b\\1\end{pmatrix}}$

And, once again, translation is not linear!

Alternatively, if you claim ${\begin{pmatrix}x\\y\\0\end{pmatrix}}$ is in the set, so we can use normal vector addition

${\begin{pmatrix}1&0&a\\0&1&b\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\0\end{pmatrix}}={\begin{pmatrix}x\\y\\0\end{pmatrix}}\neq {\begin{pmatrix}x+a\\y+b\\0\end{pmatrix}}$

So now we've got a normal matrix multiplication, hence linearity, but the matrix isn't a translation anymore! It's only a translation when we force the last co-ordinate to be "1".

And, once again, you're completely wrong! You're batting 1.000 on wrongness this! You're the Ted Williams of incorrect! The Don Bradman of not-the-right-answer! The Babe Ruth of completely-wrongheaded-nonsense! Want to play again?

-- GWO (talk)

And, once again, you're completely wrong! You're batting 1.000 on wrongness this! You're the Ted Williams of incorrect! The Don Bradman of not-the-right-answer! The Babe Ruth of completely-wrongheaded-nonsense! Want to play again?

I like this. Too bad that I understand nothing about the sport you refer to. --Lantonov (talk) 08:42, 23 May 2008 (UTC)

Oooh, I'd missed this bit

Axioms of closure:

9) ${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}\in V$

10) $a{\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}\in V$

This is genius. I love this. At exactly the crux of your definition of V, you dodge the bullet. At precisely the point where you had to choose between

${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}={\begin{pmatrix}u_{1}+v_{1}\\u_{2}+v_{2}\\1\end{pmatrix}}$ or ${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}={\begin{pmatrix}u_{1}+v_{1}\\u_{2}+v_{2}\\2\end{pmatrix}}$

you suddenly get all coy and just write. $\in V$. Similarly when you have to choose between

$\lambda {\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}={\begin{pmatrix}\lambda u_{1}\\\lambda u_{2}\\1\end{pmatrix}}$ and $\lambda {\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}={\begin{pmatrix}\lambda u_{1}\\\lambda u_{2}\\\lambda \end{pmatrix}}$

again, no detail, just a dishonest little $\in V$. You spell absolutely everything else out long hand, and when you get to the bit you can't be consistent about, you write $\in V$. That's truly magnificent.

Until now, I've assumed good faith, but I've just realised: you know you're wrong. If you didn't know you where wrong, you had provided enough detail to hang yourself right there. -- GWO (talk)

I tried to explain to you above the whole projective geometry and homogenous coordinates in a nutshell but you dismissed everything as "rambling" and evidently didn't read it, much less try to understand it. That's why you completely fail to understand what this formulae are talking to you. Well, I will try to explain for a second and I hope, last, time. You failed to see that I write homogenous coordinates in the form (x/r, y/r, r), didn't you? This is not my whim, that's how those coordinates are defined. So don't make me explain about double quotient, etc. because it will be too much ramble without anybody listening to it. Usually, I send my students with a D for such homework.

Let me see where to start from. Probably with your biggest mistake of failing to see the meaning of homogenous coordinates. Now heed this. !!!VERY IMPORTANT!!! Vector ${\begin{pmatrix}x/1\\y/1\\1\end{pmatrix}}$ is exactly the same vector as ${\begin{pmatrix}2x/2\\2y/2\\2\end{pmatrix}}$, ${\begin{pmatrix}1.355x/1.355\\1.355y/1.355\\1.355\end{pmatrix}}$, ${\begin{pmatrix}ex/e\\ey/e\\e\end{pmatrix}}$, and infinitely many other vectors. All these vectors are the same as the vector (in non-homogenous coordinates) ${\begin{pmatrix}x\\y\end{pmatrix}}$ and all these vectors are in the vector space. If the last coordinate is not 0, these vectors are the usual vectors in the Euclidian plane. If the last coordinate is 0, then the vector ${\begin{pmatrix}x/0\\y/0\\0\end{pmatrix}}$ is in the projective space, more exactly, it is built on the point at infinity (1D object). The vector ${\begin{pmatrix}x/0\\y/0\\0\end{pmatrix}}$ is in the vector space. For starters, this is enough to debunk the composition you are gloating over. I will start from the last of your statements because I do not want to read everything. Go back and correct the rest accordingly.

This is genius. I love this. At exactly the crux of your definition of V, you dodge the bullet. At precisely the point where you had to choose between

${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}={\begin{pmatrix}u_{1}+v_{1}\\u_{2}+v_{2}\\1\end{pmatrix}}$ or ${\begin{pmatrix}u_{1}\\u_{2}\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}\\v_{2}\\1\end{pmatrix}}={\begin{pmatrix}u_{1}+v_{1}\\u_{2}+v_{2}\\2\end{pmatrix}}$

you suddenly get all coy and just write. $\in V$.

I will write this more explicitly, to fill the details you missed.

${\begin{pmatrix}u_{1}/1\\u_{2}/1\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}/1\\v_{2}/1\\1\end{pmatrix}}={\begin{pmatrix}u_{1}/1+v_{1}/1\\u_{2}/1+v_{2}/1\\1\end{pmatrix}}$

is the same as

${\begin{pmatrix}u_{1}/1\\u_{2}/1\\1\end{pmatrix}}+{\begin{pmatrix}v_{1}/1\\v_{2}/1\\1\end{pmatrix}}={\begin{pmatrix}2u_{1}/2+2v_{1}/2\\2u_{2}/2+2v_{2}/2\\2\end{pmatrix}}$.

The vector ${\begin{pmatrix}2u_{1}/2+2v_{1}/2\\2u_{2}/2+2v_{2}/2\\2\end{pmatrix}}$ is in the vector space, and is exactly the same as vector ${\begin{pmatrix}u_{1}/1+v_{1}/1\\u_{2}/1+v_{2}/1\\1\end{pmatrix}}$. Got this? Now correct the rest to see if you understand it. --Lantonov (talk) 06:56, 23 May 2008 (UTC)

I think you have the normalization backwards. For example, it should be (x y 1) = (2x 2y 2). But with that, addition no longer works: (x y 1) + (x y 1) = (2 x/2 2y/2 2/2) = (x y 1). Also, you say the vector (x/0 y/0 0) is in the vector space. I assume you mean (0x 0y 0) is in the vector space; but how do you normalize that? Divide by zero? While homogeneous coordinates are tremendously useful, I don't think they are a vector space in the usual sense. After all, the entire idea is to remove the zero from the vector space to get an affine space. That is why addition and scalar multiplication don't work. The only operation that makes sense is matrix operations. That's not to say the eigenvectors of those matrix operations aren't meaningful, but it's not going to have the usual meaning after perspective division. —Ben FrantzDale (talk) 11:29, 23 May 2008 (UTC)

BenFrantzDale, welcome to this discussion. It promises to get interesting and productive. Sorry, but your additions are wrong too. Your mistake is dividing the last coordinate by two in your addition. You see, the last coordinate is not a coordinate in the usual sense. It is any real number, including 0. So to write your examples explicitly, it is:

${\begin{pmatrix}x/1\\y/1\\1\end{pmatrix}}+{\begin{pmatrix}x/1\\y/1\\1\end{pmatrix}}={\begin{pmatrix}2x/2+2x/2\\2y/2+2y/2\\2\end{pmatrix}}={\begin{pmatrix}2x/1\\2y/1\\1\end{pmatrix}}$

If you do not understand why this is so, give more examples to thrash that out.

${\begin{pmatrix}x/1\\y/1\\1\end{pmatrix}}={\begin{pmatrix}2x/1\\2y/1\\2\end{pmatrix}}$ is wrong too. It should be:

${\begin{pmatrix}x/1\\y/1\\1\end{pmatrix}}={\begin{pmatrix}2x/2\\2y/2\\2\end{pmatrix}}$

The trick is that the divisor should equal the last coordinate. The other 2 coordinates should be multiplied according to the divisor so that the balance is maintained: x = x/1 = 2*x/2 = 1000*x/1000 and so on. This is done depending on the last coordinate.

You correctly observed that ${\begin{pmatrix}x/0\\y/0\\0\end{pmatrix}}$ is not written very correctly if by x an y are understood the usual, non-homogenous coordinates. In such case, it should be written ${\begin{pmatrix}0x/0\\0y/0\\0\end{pmatrix}}$ but the geometric meaning of this vector is now different, it is not exactly the same as ${\begin{pmatrix}1x/1\\1y/1\\1\end{pmatrix}}$ because ${\begin{pmatrix}0x/0\\0y/0\\0\end{pmatrix}}$ is in projective space while ${\begin{pmatrix}1x/1\\1y/1\\1\end{pmatrix}}$ is in the Euclidean plane. Let's also observe that 0/0 is finite number, so it can be a bona fide coordinate.

After all, the entire idea is to remove the zero from the vector space to get an affine space. Exact. Objects with 0 as the third (or fourth) coordinate are the pets of projective geometry, and are defined extensively as point (1D), line (2D) and plane (3D) at infinity. Plane at infinity is only one single object and it is the whole space. Translation sends the eigenvectors in projective space and rotation sends the eigenvectors in complex space and this is why they are inherently difficult to understand. Translation has eigenvalue 1 with algebraic multiplicity 3 and geometric multiplicity 2. The two eigenvectors are ${\begin{pmatrix}1\\0\\0\end{pmatrix}}$ and ${\begin{pmatrix}0\\1\\0\end{pmatrix}}$ and are in the projective space (0 as the third coordinate). Regards. --Lantonov (talk) 12:20, 23 May 2008 (UTC)

I disagree; I maintain that in homogeneous coordinates, (x y 1) = (2x 2y 2) up to perspective division. You say that

(x y 1) = (2x/2 2y/2 2),

but if we apply perspective division to that, we get

(x y 1) = (x/2 y/2 1)

which implies x = x/2. Surly you aren't claiming that? —Ben FrantzDale (talk) 00:10, 24 May 2008 (UTC)

When one works only with finite elements, perspective is applied as you state, by multiplying the coordinates with the perspective factor. In this case, yes, we have indeed (x y 1) = (2x 2y 2) with a perspective factor 2. The difficulty comes with infinite elements when the perspective factor is 0. Then all vectors have coordinates 0 so there is no way to work with them. If instead of perspective factor, we apply perspective quotient, in this case, 2/2, then finite quotients 0/0 can be handled. We must be careful in handling the perspective quotient in this case as we have to distinguish between 2*x/2, and 1*x/1. We cannot have vectors of the type (x/2 y/2 1). All three numbers (3rd coordinate, divisor and multiplier) must be equal. --Lantonov (talk) 06:34, 26 May 2008 (UTC)

Lantonov, thanks for the reply. You've convinced me that you are talking about something I don't quite understand but that you have read and seem to understand. I may read this back over and try to understand what you are talking about. That said, I think this discussion has strayed from talk of Eigen-things. —Ben FrantzDale (talk) 00:19, 28 May 2008 (UTC)

I can explain the rationale behind it but since talk strayed from Eigens, I can explain it on my or your talk pages, if you want. The reason is not in what I said here. --Lantonov (talk) 09:04, 28 May 2008 (UTC)

Enough is enough

I commend the other editors for their inordinate patience with Lantonov, but at some point one simply has to say that he/she obviously doesn't know what he/she is talking about, and cannot comprehend the limits of his/her knowledge in this subject. There is no point in expending kilobytes of discussion on this nonsense. Wikipedia is not a debating society.

Here is a simple test: Lantonov, can you point to a reputable published source that describes your so-called vector space etc., using your terminology and definitions? If not, we will all merely point you to the WP:NOR policy every time you post.

—Steven G. Johnson (talk) 14:39, 23 May 2008 (UTC)

Steven, Ben. Thank you. I've felt like somewhat screaming in the wilderness. I am now going to consider this issue settled, and those correct those errors of Lantonov's that remain in the article. -- GWO (talk)

In the future, the WP:NOR policy is a great way to avoid at least some of these endless debates (and was originally designed for just such a purpose). Just ask them for a reference supporting their novel interpretation/theory/definition/notation/etc., and if they can't provide one, then you can dismiss the issue without further argument. —Steven G. Johnson (talk) 21:59, 23 May 2008 (UTC)

Yes, I can point to a reputable published source that describes "my so-called vector space". It is my textbook in analytic geometry, which I studied in my second year of undegradute study in Mathematics in Sofia University about 30 years ago. It is: Petkanchin, Boyan (1966) Analytic Geometry. Science and Arts Publishers, 3rd edition, Sofia, 846 pp. Unfortunately, this book is not with me at the moment. Tomorrow I will take it from my home and provide you with exact locations and citations from it, should you need them. About the reputabilty of Academician Prof. Dr. Boyan Petkanchin, see the article in Bulgarian Wiki bg:Боян Петканчин. There, this book is listed as Аналитична геометрия. Наука и изкуство, С, 1952, 807 с; II изд. 1961, 843 с, III изд., 1966, 846 с. Note that most of my contributions in this article are provided with a source, and the larger parts of references and inline citations here are provided by me, including proofs of theorems in Wiki books. --Lantonov (talk) 14:04, 26 May 2008 (UTC)

The book is with me now. The axioms for the vector space are on pp. 330-331. --Lantonov (talk) 06:59, 27 May 2008 (UTC)

How convenient that it is impossible for almost anyone on the English Wikipedia to read this book and check that it backs you up, or whether you are fundamentally misunderstanding it (or mistranslating its terminology into English). We have to insist that you provide a reputable reference in English. If this is a textbook definition as you claim, then you should be able to find a widely available English textbook that has the same material. —Steven G. Johnson (talk) 02:33, 28 May 2008 (UTC)

One of the reasons I did not include this material in the article was that I was unable to find an English textbook on it. Another reason was that it is too old. I do not know who are the "we" and on what grounds they are insisting but I have to point out that there is no requirement in the guideline for scientific references that the text be in English. The most important requirement is that the text be academic, and the book that I point to is purely academic as you can see it in the Sofia Universtity curricula up until the mid 1990s when it was replaced by more modern textbooks. I can translate relevant parts of the text verbatim but given the mistrust with which my contributions are met here, I doubt it is worth the effort. Alternatively, I can scan the relevant pages and send them to anyone interested; however, I am not sure if this infringes the author's copyright. I see no point in it anyway, as most of my text, sourced from widely available English books and provided with inline references, was deleted. Still, I will look for English sources and will give them here when I find some. I have 3 sources in English that describe the general properties of projective spaces but none of them explicitly includes the defining axioms as the above textbook does. For such general description, you can look, of course, in the article Projective space and references therein. I can give you, however, an English book that says verbatim: "in the homogenous coordinates the translation is a linear transformation". It is: Treil, Sergei. 2004. Linear Algebra Done Wrong. Brown University Press. Sergei Treil is from the Department of Mathematics at Brown University. The cited text is on p. 33. When you read it, you will see that there is nothing wrong in it, in spite of its title which is obviously a calque of the Schaum series book: Linear Algebra Done Right. However, if you tell me the opposite statement: "In homogenous coordinates translation is NOT linear transformation", I will believe you and will not argue about it. I will accept that I "fundamentally misunderstand" something. Obviously, translation cannot be both linear, and non-linear. Anyway, things may be more complicated and you may not want to explain them to me because you think they are beyond my ability to comprehend.--Lantonov (talk) 05:35, 28 May 2008 (UTC)

I was wrong in saying that Petkanchin was discontinued in mid 1990s. See thе official curriculum from Sofia University for 2005 [1] and (in English) in 2008 [2] for Analytical Geometry where 1. Петканчин. Аналитична геометрия. 1966. София and 1. B. Petkanchin, Analytical Geometry, Nauka & Izkustvo, Sofia, 1966 (in Bulgarian) is still present as a textbook. --Lantonov (talk) 09:55, 28 May 2008 (UTC)

Mention of generalized eigenvectors

There is already an article about generalized eigenvectors, but should we include a short section? Or at least mention them and link to the other page? daviddoria (talk) 21:27, 16 September 2008 (UTC)

External Links

I wish to add an external link to this and other related articles. Please see * Talking Picture Book version - of this article. It is just another way of presenting information. It is completely free to the user. I have spent a lot of time creating these articles. Maybe after my death, I will get a reward for my hard work, but in the meantime it could be of benefit to the Wiki users/donors. Wayp123 (talk) 22:55, 7 December 2008 (UTC)

Can I add my website here, since it did help me to make a contribution to this article. It did make my life easier, one of wiki's slogans, it may help someone else. It is not considered advertising or self promotion if I can get permission here. It can be a neutral link so I can put links directly to the download of the file and the program without going to the website. Downloads are allowed if it is stated in the link. It is not advertising since I am not competing with other websites. Wayp123 (talk) 22:01, 9 December 2008 (UTC)

Consensus at Wikipedia talk:WikiProject Mathematics#Some external links is that it is not appropriate to add links to your web-site to Wikipedia articles. Gandalf61 (talk) 09:17, 10 December 2008 (UTC)

A learning curve too steep.

I've tried to respond to the request for a less confusing lede.

I find the picture of Mona Lisa singularly unhelpful, because the eye sees the picture as turned in three dimensions rather than skewed in two dimensions. Can anyone provide a better picture to illustrate this article?

Rick Norwood (talk) 13:44, 25 July 2009 (UTC)

I edited the caption rather than the picture in the hope of clarifying that the picture is simply being sheared vertically in the plane rather than turned in three dimensions. --Vaughan Pratt (talk) 09:46, 7 January 2010 (UTC)

is waaaaay too hard :'( I feel like going emo... I HATE ALL THIS MATHSS, brings tears to my eyes. I appreciate the COLOR picture in this article though (of mona lisa.. which is better than some mathy plane stuff).

Browser issues

The six animations of the girders in the Vibrations section really slow down my browser when it has to display them - Safari 4 on a 2.4 GHz Core 2 Duo machine. Maybe these should be removed to the appropriate page rather than bloating up this one. Brammers (talk) 19:53, 14 October 2009 (UTC)

I've removed 5 of the 6. They can be found on the vibration article. I don't think animated gifs are the way to go with this. It's better to do movies that the user can explicitly decide to play if they so choose. --Rajah (talk) 09:38, 17 November 2009 (UTC)

Explaination

The layman explanation provided both in the lead and "technical definition" section proved rather helpful, particularly given the demands to be encyclopaedic. Congratulations. - Jarry1250 ^{[Humorous? Discuss.]} 20:59, 28 October 2009 (UTC)

Characteristic Polynomial

What does:

"Proving the aforementioned relation of eigenvalues and solutions of the characteristic equation requires some linear algebra, specifically the notion of linearly independent vectors: briefly, the eigenvalue equation for a matrix A can be expressed as"

mean exactly. I mean what exactly is this aforementioned relation? Why not just state exactly what is meant here? Why is the linear algebra needed? Explain what is being done. — Preceding unsigned comment added by 68.147.37.61 (talk) 15:05, 18 October 2012 (UTC)

New lede

I disagree with the new lede. My objections are:

1. It is too long and too detailed - see WP:LEDE.
2. It is not correct to say that the concept of egenvalues and eigenvectors is only defined for vector spaces over an algebraically closed field. Eigenvalues and eigenvectors make perfect sense for vector spaces over the reals - the only proviso is that some linear transformations (such as rotations) may not have the expected full complement of eigenvectors. This situation is discussed in the Existence and multiplicity of eigenvalues section.
3. In a complex vector space, multiplication by a complex number scalar is not rotation. So it is incorrect to say that multiplying (i,1) by i rotates it 90 degrees clockwise. In a complex vector space, (i,1) and (-1,i) are parallel vectors.

I propose reverting to the previous lede, and then considering incremental improvements instead of a complete rewrite. Gandalf61 (talk) 11:38, 7 January 2010 (UTC)

Okay, I see Rick Norwood has now reverted to the previous lede, which is fine with me. Gandalf61 (talk) 13:24, 7 January 2010 (UTC)

Fine by me too. I just wanted to try out a more detailed lede to get people's reactions. Now that people have had a chance to compare the two, by way of justification of my reasoning behind my suggested replacement let me address the points you raise in reverse order (sorry about the less neat indenting reversal seems to entail).

3. The rotation issue revolves around the question of whether one considers Cⁿ as an n-dimensional space over C or a 2n dimensional space over R. As you point out these aren't entirely equivalent, yet they have enough in common, e.g. the same Euclidean metric affinely and the Fubini-Study metric projectively, that the latter perspective can be quite helpful in visualizing what's going on. Insistence on the former to the exclusion of the latter deprives the student of the visual benefit available via the natural metric when grappling with the difficult notion of how ostensibly linear behavior can nevertheless have an oscillatory character, as well as leaving them unprepared to cope with the latter view when encountered in the literature, e.g. the Bloch sphere which Wikipedia explains rotationally rather than worrying about "parallel vectors."

2. While it's true that every matrix over F has the property that its characteristic function factors linearly over some (possibly improper) extension of F which need not be algebraically closed (the Fibonacci example being a case in point, factoring over R as (x-φ)(x-ψ)), from a pedagogical standpoint this is a level of sophistication that I would be inclined to dispense with in the lede by starting out with algebraically closed fields. (If your thinking was that one might be able to teach eigenvectors before complex numbers, I confess that possibility had not occurred to me.) [On second thoughts it would be simpler, as well as addressing your concern about accuracy, just to drop any mention of the field in the first sentence, other than something like "typically over the reals or complex numbers."]

1. Regarding WP:LEDE, the essential feature of a lede is that it stand alone as a concise overview of the article summarizing the most important points, which I felt the present lede was falling down badly on. My replacement is too long but I wrote it that way on the principle that it is easier to maintain the coherence of a lede by deleting excess detail than by allowing it to grow like Topsy in order to reflect the article's content. Three relevant datapoints are that this article is 61 kilobytes while the hyperbola article is only 44 kilobytes, yet prior to December 2008 the lede of the latter had grown to 10% longer than my proposed lede for this article. Its present lede is half the old length, achieved by rearranging the scattered thoughts of the old one more efficiently.

The substance of the present lede resides in the first five sentences of the second paragraph, which gives an appealingly visual picture of the three concepts in the title, but at a leisurely speed that I would have thought was more appropriate to the body of the article where there is more room for this sort of pedagogy. The immediately preceding two sentences state that the three concepts can be computed and give a (very partial) list of applications, namely information about a matrix, matrix factorization, finance, and QM. A sentence is devoted to the etymology of "eigen" without however making clear how that makes it one of "the most important points." The remaining five sentences of the lede concern themselves in various ways with the prerequisites and their relevance, which one would expect could be combined without loss of clarity in a single sentence. For example the first and third sentences state In mathematics, eigenvalue, eigenvector, and eigenspace are related concepts in the field of linear algebra. Linear algebra studies linear transformations, which are represented by matrices acting on vectors, while the last two sentences say this again with a double negative, The concepts cannot be formally defined without prerequisites, including an understanding of matrices, vectors, and linear transformations. The technical details are given below. (The computability of the concepts is also promised "by a method described below.")

It seemed to me that any attempt to fix these problems incrementally could only lead to the sort of growth that happened with the old hyperbola lede, and that it was better to start from a coherent account of "the most important points" using a generous definition of "important" and then gradually shorten it by progressively raising the bar on "important" until the length was agreed to be within WP:LEDE's guidelines while still doing justice to the complexity of this subject as measured by its 61 kb size. I don't believe the present lede meets that last objective. --Vaughan Pratt (talk) 21:26, 7 January 2010 (UTC)

In general, the lede to any article should be readable by any literate reader, which means the lede will necessarily lack the compression usually found in writing for mathematicians. The lede should be mathematically correct, but non-technical. The two sentences you compare in point 2 do not say the same thing. The first places the article in its context (linear algebra). The second explains that the ideas are sufficiently technical that they cannot be precisely defined without prerequisites. Rick Norwood (talk) 13:15, 8 January 2010 (UTC)

Basis of Eigenvectors

One thing I missed in the article is mention of a basis of eigenvectors.

I know that an orthonormal basis van be constructed for any hermitian matrix consisting only of the eigenvectors of the matrix. I think the article would benefit from some information about whether the eigenvectors span the entire space (and thus can be used for a basis, even if it's not orthonormal) and any requirements for the existence of such a basis. —Preceding unsigned comment added by 131.155.59.211 (talk) 11:14, 5 February 2010 (UTC)

Example

"If a matrix is a diagonal matrix, then its eigenvalues are the numbers on the diagonal and its eigenvectors are basis vectors to which those numbers refer."

Does it need to specify that the basis vectors in the example must belong to the natural basis, i.e. (1,0) and (0,1), rather than just any basis? Dependent Variable (talk) 11:10, 8 February 2010 (UTC)

Yes, it would be with respect to the natural basis. Plastikspork _―Œ^(talk) 17:21, 9 February 2010 (UTC)

Cut to the chase?

I'll resist the urge to tamper, as I'm only a learner, but I think it would be good to put the formal definition (the one in the box in "Technical definition") at the head of the article, or very near it. Or, if need be, have a very brief informal, verbal definition in the intro then follow that immediately with a single definition section based on "Technical definition", before any History or Example(s) or background material on related concepts. If that isn't acceptable, maybe put the formal definition in place of the Mona Lisa? It seems a shame that such a clear and simple definition of what an eigenvector actually is should be hidden away so far into the article, after repeated vaguer, meandering, less rigorous definitions and incidental historical detail. I remember thinking when I learnt the meaning of the term eigenvector elsewhere, "Oh is that all, why didn't the Wikipedia article just say that?"

There are already articles on vectors (e.g. Vector space), matrices, linear transformations, etc., so attempts to define each of these concepts seems an unnessecary distraction; a link should suffice. I don't think we need any of "Mathematical definition" (and how is the technical definition any less mathematical?), except the list of applications (which can go elsewhere). Dependent Variable (talk) 11:09, 12 February 2010 (UTC)

Shear Eigenvector solution equation

The shear eigenvector solution equation has the shear factor, k, negated. This was changed on 14 June 2008 at 08:14. Can anyone explain this, as it doesn't match the other examples? 20.133.0.8 (talk) 10:10, 12 March 2010 (UTC)

The derivation of the eigenvector was solving $A\mathbf {x} =\lambda \mathbf {x}$ by rewriting it as $[\lambda I-A]\mathbf {x} =\mathbf {0}$, hence the k in the upper right hand entry of A turned into -k. I have rewritten the derivation to make it simpler and clearer. Gandalf61 (talk) 10:52, 12 March 2010 (UTC)

Rotation solution equation

I don't follow this example, it would seem to me that the eigen vectors and values are a function of the angle of rotation. The example seems correct for a rotation of pi/4 rad or 45 degrees but the example appears parametric and not specificJshriver2 (talk) 23:45, 26 March 2011 (UTC).

Symmetric versus other matrices?

I completely undertstand eigendecomposition in the context of symmetric positive-definite matricies. That the eigenvectors describe the principal directions of things like a shear deformation or a covariance matrix makes perfect sense (although I still don't understand why so many introductions to eigenvectors make it so complicated; looking at a gaussian ellipsiod in n-dimensions it is obvious by inspection what the eigenvectors are and what they mean!)

What about other matrices? There, the intuition is less clear and I still don't have a firm grasp. Symmetric indefinite or negative-definitie eigenvectors make sense: the just squash things completely in one direction, or inverting things, but what about asymmetric matrices? As I understand it, all other matrices produce complex eigenvalues and eigenvectors. I can see algebraically where complex eigenvalues and eigenvectors come from, and that they "work". However, while I have some intuition for complex numbers, I really don't for complex vector spaces. I understand that the all-real eigenvalue-eigenvector pair of a rotation matrix in R³ corresponds to the rotation axis, and that in R² the rotation axis has to be complex in that it somehow has to be orthogonal to R².

So:

1. What is the intuition for the meaning of the eigenvectors and -values of non-symmetric matrices?
2. Is there a taxonomy of eigenspaces beyond "Hermitian<=>orthogonal and real, non-Hermitian <=> complex, not necessarily orthogonal"?

As for the article, with the exception of rotation matrices, I think everything else talks about spectra of self-adjoint operators. Since eigenvalues and eigenvectors of self-adjoint operators are (a) so common in science and engineering and (b) make so much physical sense (their directions, the fact that the directions are orthogonal, etc.), I think the article would be more approachable if it started with the particular case of Hermitian matrices and only then went on to explain the details for other cases. —Ben FrantzDale (talk) 14:48, 19 April 2010 (UTC)

Eigenbasis redirect

If "Eigenbasis" is going to redirect here, then this article should say something about it. Bender2k14 (talk) 18:12, 19 September 2010 (UTC)

Commuting Hermitian matrices have same eigenvalues

I came across a thoerem stating that two commuting Hermitian matrices always have same eigenvalues, but I could not find on Wikipedia. This is critical to quantum mechanics because it is used to show that commuting dynamic variables can be measured simultaneously while non-commuting cannot. Can anyone write something about this theorem and prove it?--Netheril96 (talk) 09:43, 7 October 2010 (UTC)

Silly title

The title should not be "Eigenvalue, Eigenvector and Eigenspace." When we title articles on Wikipedia, we do not name a bunch of concepts we cover in the article: that is for the headers of the table of contents. Instead we title it after the overarching concept we wish to describe in the article. The concepts of "eigenvector" and "eigenspace" are defined in terms of the definition of eigenvalue, so it makes aesthetic and logical sense to consider them as part of the overarching theory of eigenvalues.

Defn: An eigenvalue of a linear operator T on a vector space V is a number a such that Tv=av f.s. nonzero v in V.
Defn: Any vector v which satisfies Tv=av for some eigenvalue a is called an eigenvector.
Defn: The set of vectors v which satisfy Tv=av for some eigenvalue a is called the eigenspace of a.

(one would then prove that an eigenspace forms a subspace of V, so it makes sense to call it a "space")

You see that the definitions of eigenspace and eigenvector are dependent on the overarching concept of eigenvalue. The article should be called Eigenvalue, and "Eigenvector" and "Eigenspace" should redirect here.MarcelB612 (talk) 21:15, 28 October 2010 (UTC)

In general, I agree. (I at least want a serial comma if we leave it roughly as-is.) What about Eigenanalysis (which presently redirects to this page)? That seems to describe well the overarching concept as a whole without preference for one of the parts. —Ben FrantzDale (talk) 13:39, 29 October 2010 (UTC)

I agree that the title is silly. But I would prefer Eigenvalues and eigenvectors (plural, as there may be more than one of them for a given matrix). No need to add "eigenspaces". I see eigenvalues and eigenvectors as two distinct concepts. We first compute eigenvalues, then eigenvectors, but this does does not mean that eigenvector is a concept contained in the definition of eigenvalue.

Eigenspace, however, is clearly redundant with respect to "eigenvectors" (plural), as much as spectrum is redundant with respect to "eigenvalues" (plural).

— Paolo.dL (talk) 16:34, 7 December 2010 (UTC)

With a very small range of exceptions, article names in Wikipedia should almost always be singular, not plural - see WP:SINGULAR. Gandalf61 (talk) 16:42, 7 December 2010 (UTC)

I see. Thank you for the info. However, I believe that in this case the title would stand for an abbreviation of: "Eigenvalues and eigenvectors of a matrix". This is why I would use the plural. There is a set of eigenvalues and eigenvectors for each matrix. This is similar to Arabic numbers. Plural names of Classes are allowed by WP:SINGULAR. — Paolo.dL (talk) 17:25, 7 December 2010 (UTC)

The title does not seem silly at all to me. It is a natural title for three closely related topics. Splitting the article into three would require needless repetition. A quick search turns up articles titled Tic-Tac-Toe, Huey, Dewey, and Louie, and Faith, Hope and Charity. I'm sure there are many more articles with three word titles. Rick Norwood (talk) 17:57, 7 December 2010 (UTC)

You are right that the title is not silly. It just appears questionable in the opinion of some editors, including me. But for sure it would be silly to maintain that the title is too long just because it contains three names. And up to now nobody maintained that.

The problem is not at all in the number of words. There are many terms related to eignevalues and eigenvectors, and I am sure you agree that we cannot list all of them. Someone calls them informally eigenstuff. The problem is to select the main ones. Eigenspace (a set of eigenvectors) is only one of the related terms. A very similar concept is spectrum (a set of eigenvalues)... Hence, a title longer than the current one, that contained the term spectrum as well, would seem to me less questionable than the current one. Therefore, a list of articles with long titles is not necessary. We are more interested in articles or book chapters about "eigenstuff". I found seven of them (see below), none of which included the term "eigenspace".

Also, nobody suggested to split the article in two or three parts. On the contrary, we all maintained that the article should stay as it is.

— Paolo.dL (talk) 10:55, 8 December 2010 (UTC)

References

For instance, see:

• Sharipov R.A. (1996). Course of Linear Algebra and Multidimensional Geometry. P. 66: Eigenvalues and eigenvectors
• Kuttler K. (2007). An introduction to linear algebra. P. 159: 7. Spectral theory - 7.1 Eigenvalues and eigenvectors of a matrix
• Beezer R.A. (2006). A first course in linear algebra - Section EE: Eigenvalues and eigenvectors
• Anthony Carter T., Tapia R.A., Papaconstantinou A. (2003) Linear Algebra. Chapter 9: Eigenvalues and eigenvectors
• From SOSMATH: Eigenvalues and eigenvectors
• From HMC: Eigenvalues and eigenvectors
• From CNX: Eigenvectors and eigenvalues

Also, there's another book chapter:

• Aldrich J. (2006). Eigenvalue, eigenfunction, eigenvector, and related terms. In Miller J. (Ed.). Earliest Known Uses of Some of the Words of Mathematics.

but in Wikipedia we have a separate article for eigenfunction and related terms such as eigenmode and eigenface, so this title is not appropriate to this article.

Notice that in Wolfram MathWorld there are two articles: Eigenvector, and Eigenvalue. But I prefer a single article, as this one.

— Paolo.dL (talk) 17:25, 7 December 2010 (UTC)

List of options

I added some references above. Notice that now we have seven articles or book chapters/sections, and none of them includes in its title the term "eigenspace". In short, four options were suggested above and a fifth one found in the literature:

1. Eigenvalue, Eigenvector and Eigenspace (current title; only 1 contributor out of 5 likes it)
2. Eigenvalue (proposed by 1 contributor)
3. Eigenanalysis (proposed by 1 contributor; some contributors believe it does not meet WP:COMMONNAME)
4. Eigenvalues and eigenvectors (most common in the literature and proposed by 1 contributor - see references above)
5. Spectral theory (see chapter title by Kuttler K. above; no contributor suggested this title; we think it does not meet WP:COMMONNAME)

Up to now, I believe we must temporarily conclude that the current title should be changed (3 contributors in favor, 1 against), and the title should be Eigenvalues and eigenvectors (most common in the literature). Everybody agrees that this article should not be split in two separate articles ("Eigenvalue", and "Eigenvector"). — Paolo.dL (talk) 10:55, 8 December 2010 (UTC)

Call for comments or motivated vote

Please vote (but please first read the previous comments). I propose to wait no more than 10 days. Then we'll take a final decision (this discussion started about 2 months ago). — Paolo.dL (talk) 10:55, 8 December 2010 (UTC)

Not really comfortable with this whole voting thing - see WP:VOTE - but if you are asking for opinions then I am in favour of 1 or 2 (assuming that 2 includes a redirect from Eigenvector and Eigenspace to Eigenvalue) Gandalf61 (talk) 13:14, 8 December 2010 (UTC)

I apologize, Gandalf61. You are perfectly right (and your links to WP are welcome.) Actually, I did not mean to ask for a plain vote. I gave a list of options, but I assumed people would first read the contributions above, then explain their decision. I would also appreciate other suggestions and references. Probably, I should not have used the word "vote", as it may be interpreted as a plain selection from the list of options, without reference to the previous discussion. I changed the title of this section, to discourage not motivated votes.

By the way, I would like to know the reason why you selected 1 or 2, as I explained above the reason why I am against these two options. Also, please everybody provide an advice about what is the importance we should give to the references, in this case. — Paolo.dL (talk) 14:19, 8 December 2010 (UTC)

Okay, my reasons are that 4 does not conform to WP:SINGULAR, whereas 3 and 5 are less common terms and so do not meet WP:COMMONNAME. So that leaves 1 or 2. Gandalf61 (talk) 14:43, 8 December 2010 (UTC)

As I wrote above, I disagree that 4 is against WP:SINGULAR. There is typically more than one eigenvalue-eigenvector pair for each matrix. There may be as many pairs as basis vectors or Cartesian coordinates in the space upon which the matrix acts. I am sure you agree that this is the reason why almost everybody uses 4 in the list of references provided above. Thus, "Eigenvalues and eigenvectors" is very similar to Polar coordinates, which is listed among the exceptions to WP:SINGULAR. Paolo.dL (talk) 15:21, 8 December 2010 (UTC)

Yes, I know that you disagree. I, in turn, disagree with your disagreement. So there we have it. You asked for my reasons, and I gave them - I didn't expect the Spanish inquisition. We can all read your arguments; there is no need to repeat them ad nauseum. Having asked for other editors' opinions, it would be better if you waited patiently to see if a consensus emerges, instead of cross-examining every contributor who does not support your favoured option. Gandalf61 (talk) 15:49, 8 December 2010 (UTC)

You are free not to answer, but I am entitled to ask, when my comments are ignored. I wish to reach an agreement. An agreement can be reached only if opposite opinions are explained. Inappropriate reference to Wikipedia policies may heavily bias the discussion, and I am not going to let it happen. Paolo.dL (talk) 16:15, 8 December 2010 (UTC)

I didn't ignore your comments; I read them, thought about them and decided I didn't agree with them. And I don't see how my reference to the Wikipedia:Article titles policy in a discussion about an article's title can possibly be "inappropriate". We may have different ideas about how to interpret that policy, but this article's title must eventually conform to it one way to another. You can't just decide that a policy is "inappropriate" because you don't agree with it - that's not how we do things on Wikipedia. Gandalf61 (talk) 13:16, 9 December 2010 (UTC)

You clearly did not read my comments with attention. I never wrote that the "policy is inappropriate" because I "don't agree with it"! I wrote that in my opinion, according to WP:PLURAL#Exceptions, WP:SINGULAR (which is just a two-row summary of the main article WP:PLURAL) does not apply in this case. More exactly, the "general" part of the policy does not apply, but its exceptions, which are part of it, do apply. And I explained why. In turn, you kept referring to the general part without explaining why, and this is in my opinion inappropriate. Not the policy. You just wrote "I disagree with your disagreement". Is that "how we do things on Wikipedia"? Would you mind to help us to reach an agreement, please? Thank you. — Paolo.dL (talk) 14:05, 9 December 2010 (UTC)

We don't need to reach agreement - that is not a realistic goal. We just need to determine consensus and then agree to abide by it (well, I suppose that is an agreement of sorts). Anyway, you have my opinion and my reasons and my agreement to abide by the consensus, and that's all you are getting from me. I am done here. Gandalf61 (talk) 16:45, 9 December 2010 (UTC)

I agree that we should look for consensus rather than take a vote. While I prefer the old title, I have no strong objection to "Eigenvalue and eigenvector". Keep in mind that whoever makes the change, if there is one, must use "what links here" to change all of the articles that link to "Eigenvalue, eigenvector and eigenspace" so that they link to the new title, to avoid redirects. There are hundreds of articles involved, with a convoluted path of redirects already in place. That person must also provide links so someone searching under the old title, or a varient of the old title, or a part of the old title, finds their way here. One advantage of the shorter title is that it avoids the question of whether or not to include an Oxford comma. If we do decide to make the change, there is so much work involved that we had better be very sure there is a consensus, and nobody is likely to come along an change it back. I certainly won't. Rick Norwood (talk) 14:52, 8 December 2010 (UTC)

Thank you for your advice. Did you choose the singular form "Eigenvalue and eigenvector" because of WP:SINGULAR? In my opinion, WP:PLURAL#Exceptions clearly says that in this case WP:SINGULAR does not apply. Please read my previous comment and let me know. Paolo.dL (talk) 15:35, 8 December 2010 (UTC)

I chose the singular because the current title uses singulars. I have no preference one way or the other. For what it is worth, I would rather see this article restored to Featured Article status than see the name changed. Rick Norwood (talk) 16:18, 9 December 2010 (UTC)

I like "Eigenvalues and eigenvectors" because it is used often for textbook chapters. —Quantling (talk | contribs) 18:03, 8 December 2010 (UTC)

Is there a strong consensus on what the new title should be? I don't see it. Looks like one vote each for a wide range of new titles. Before we make a major change like this, I think we need a consensus favoring a particular title. Rick Norwood (talk) 13:29, 11 December 2010 (UTC)

Not totally true. Actually, option 4 ("Eigenvalues and eigenvectors") has 2 votes out of 6, and the only editor who expressed an opinion against it wrote he will abide to consensus. On the other hand, 4 editors are against the current title. But besides our votes, shouldn't we also consider the literature? The list of articles and textbook chapters provided above sends a crystal clear message. In their titles:

1. the term "eigenspace" (included in current title) is never used. Which supports the proposal to change the title.
2. the expression "Eigenvalues and eigenvectors" is always used, always in plural form, sometimes in reverse order, sometimes followed by the specification "... of a matrix" (this specification is always implied, even when omitted, and explains why the plural form is appropriate). The only exception is not relevant, as it splits the contents of this article in two parts ("Eigenvectors", and "Eigenvalues"), while we prefer to keep everything in a single article (you gave valid reasons for this choice).

So, at least we can say that there are strong reasons to change the current title. Not only because 4 editors out of 6 support the proposal, but also and more importantly because the word "eigenspace" (included in the current title) is never used in the above-listed titles. Moreover, options 3 and 5 can be discarded because of WP:COMMONNAME.

— Paolo.dL (talk) 16:08, 11 December 2010 (UTC)

I'm not sure 2 votes is a strong consensus, but I have no objection to "Eigenvalues and eigenvectors", provided there is a consensus. The next question would then be, who takes on the big job of making the changes needed? I would strongly oppose a change that resulted in hundreds of redirects. Rick Norwood (talk) 12:41, 12 December 2010 (UTC)

Don't worry, I confirm that 2 votes out of 6 is absolutely not a strong consensus. I only wrote that there are "strong reasons to change the current title". In other words, we know what we don't want, but we still don't exactly know what we want. But again, we shouldn't forget the literature. In this uncertain situation, (in which the only certainty is that we don't like the current title) we might decide that the wisest decision is not avoiding a decision and staying unhappy, but using the most common title in the literature. If we do, we will greatly improve the situation (four people are strongly against the current title, only one is against the title used in the literature). Using a mathemtical metaphor, I suggest to move along the steepest gradient to find minimum possible unhappiness, knowing that no solution will make everybody happy. If we do it, we will reach a stable situation, as I am sure that every other suggestion would receive a much more heated opposition. And don't worry about the redirects. I will fix them.

In other words, we don't have a strong consensus, but in the literature there seem to be a very strong consensus, and I am not the only editor who suggested to keep it into account (see comment by Quantling above).

— Paolo.dL (talk) 15:39, 12 December 2010 (UTC)

I think there is a consensus in the literature, whether or not there is a consensus among Wikipedia editors. "Eigenvalue, eigenvector, and eigenspace" is an awful title any way you go about it. First of all, the words "eigenvalue" and "eigenvector" are always plural in the literature, and the term "eigenspace" is simply derivative of "eigenvector." I would argue that so is eigenvalue, but there are multiple ways to define the concepts so the two are, admittedly separate. I believe initially suggested we term this page "eigenvalue" or "eigenvectors (one or the other), but "eigenvalues and eigenvectors" is probably even better. What isn't any good is the current title. There is really no excuse for it, it's completely out of place and it kind of makes Wikipedia look like it's out of touch. "Spectral theory" would be a fine name as well, if people wanted to go with that, although I think "eigenvalues and eigenvectors" would be more familiar to most people. MarcelB612 (talk) 00:21, 14 December 2010 (UTC)

I support renaming this page to "eigenvalues and eigenvectors" (or "eigenvectors and eigenvalues"). That covers the bases and is more approachable than "spectral theory" or "eigenanalysis". —Ben FrantzDale (talk) 00:40, 14 December 2010 (UTC)

In sum, 5 out of 6 editors agree to call this article "Eigenvalues and eigenvectors", the sixth wrote he will abide to consensus. The literature shows almost unanimous consensus. We will never be able to reach a smaller amount of "unhappiness". Because of the clear trend in the literature, any other title would be very likely to sound inappropriate to a greater amount of editors. Moreover, 4 editors are strongly against the current title. Paolo.dL (talk) 12:42, 14 December 2010 (UTC)

So, who volunteers to bell the cat? Rick Norwood (talk) 14:03, 14 December 2010 (UTC)

I can start fixing the internal links on Sunday, but in the meantime, anybody can do the change. Paolo.dL (talk) 15:49, 14 December 2010 (UTC)

I strongly suggest the person willing to do the work be the person who makes the change. Rick Norwood (talk) 13:23, 15 December 2010 (UTC)

Why this article lost its Featured Article status

To find out why this article lost its Featured Article status, go to the top of this page, locate the line that says "Article milestones" and click on the word "show" at the end of that line. Rick Norwood (talk) 14:25, 10 December 2010 (UTC)

I did what I could to improve the article. I hope this will help. Feel free to tweak my edits. Paolo.dL (talk) 18:34, 15 December 2010 (UTC)

Now, let's see if we can make this a Featured Article again.

A couple of quick thoughts before I go to give my Complex Analysis final. The picture of Mona Lisa really doesn't work -- it is hard to see that the vector has changed direction, and the human eye sees the picture as just turned to the side rather than stretched. And, below the TOC, the statement that you can define a linear transformation for a single vector is clearly a problem. More later. Rick Norwood (talk) 20:34, 15 December 2010 (UTC)

Please be careful when you edit the "Overview" section. When I read it the first time, I thought it was beautifully simple. The editor who wrote it knew how to explain math to laymen. This is much more difficult than the formal explanation. And typically, we can better understand (and hence better write) a formal explanation. I tied to respect this style when I edited. The formal definition follows in the following section. Paolo.dL (talk) 21:39, 15 December 2010 (UTC)

I think we're getting there. But now that Paolo brings up the complex numbers, we need to qualify this business about length and direction. Rick Norwood (talk) 22:04, 15 December 2010 (UTC)

No, please! I have removed my stupid sentence about complex eigenvalues. I realized it was a mistake. The intro was much more fluent and more easily readable before my edit. Resist to the temptation to make an introduction for mathematicians. Mathematicians do not need an introduction! They need details in specific sections. We write the intro for people who doesn't want to learn everything by reading it. They just want the main ideas, and they get mixed up if you add too many details. The intro is very good now. You already did a great job. Paolo.dL (talk) 22:46, 15 December 2010 (UTC)

Thanks. We do need to be careful not to say anything that is actually mathematically incorrect. Rick Norwood (talk) 12:48, 16 December 2010 (UTC)

I agree.

I am not sure it was a good idea to remove reference, in the first sentence of the intro, to the concept that the eigenvector "either keeps or reverses its direction". I guess that this is not true when the eigenvalues are complex with non-null imaginary part, but everywhere else, in the article, there's a repeated reference to length and direction. And this is how eigenvectors are always explained as far as I can remember. So, I would suggest not to worry about complex space, and only describe real eigenvectors in the first sentence of the introduction. I suggest to think about the complex space as something we cannot explain in a gentle introduction. This means it is also inappropriate and counterproductive to specify explicitly (unless it is in a footnote) that this is true only in real space. Most people doesn't care about complex space, unless they are mathematicians, and if they are mathematicians, then they don't need the informal definition, and will understand the formal definition(s), either in the intro, or in the specific sections following the intro.

Another thing I do not like is that we first speak of a matrix, then we generalize to linear transformation, but we do not say anymore that A is typically a matrix. I wrote it initially, but my sentence was deleted by someone who perhaps does not understand that the readers are not supposed to know that a matrix represents something that we call linear transformation.

Moreover, what kind of linear transformation cannot be described by a matrix? Are we sure we are really interested in exceptions? Why can't we just say A is a matrix? Eigenfunctions and other generalizations are described in separate articles.

In sum, I maintain that we need to simplify the introduction. This approach ("gentle introduction") is important to obtain the "Featured article" status. I won't have the time to fight for implementing and defending this approach. I won't be able to help you further. This is probably my last contribution.

— Paolo.dL (talk) 14:54, 16 December 2010 (UTC)

The objection to "direction" came from Sławomir Biały with the comment that most eigenvalues are complex numbers. I'm not sure in what sense this is true. It seems to me that it is true in some areas but not in others, and that most introductory texts begin with real vector spaces.

Sorry to lose you, Paolo. Did you take care of all the redirects?

Sławomir Biały, how strongly do you object to including the idea of a vector having a direction to help non-mathematicians undersand the concept?

Rick Norwood (talk) 19:08, 16 December 2010 (UTC)

I fixed all redirects after changing the title of the article. Good luck. Paolo.dL (talk) 20:43, 16 December 2010 (UTC)

Linear transformations

Every linear transformation of a finite dimensional vector space can be represented by a matrix, but the matrix is not the same as the transformation, because the matrix depends on an arbitrary choice of a basis, while the transformation is independent of the choice of a basis. This is a point that students often struggle with, and so it is important that we not confuse the two. It is ok to begin with a matrix as an example of a linear transformation, but it is not ok to give the impression that the matrix "is" the linear transformation.

As for the notation Ax (or, better, Av), let A be the differential operator D and let v be the polynomial function f(x) = 3x + 5. Then Dv = f '(x) = 3. But the matrix representation of the differential operator is an infinite matrix, and it is unclear in what sense an infinite matrix is square. Rick Norwood (talk) 13:20, 18 December 2010 (UTC)

What do you mean exactly? If I want a rotation by 90° about the x axis, whatever is the x asis, I always use the same matrix. However, I never wrote, nor imply nor gave the impression that "the matrix is the linear transformation". I wrote that "the multiplication by A" (not A) was an example of linear transformation (this info is lost in your version). On the other hand, your latest changes seem to be exactly all what is needed to give the impression that "the matrix is the linear transformation". So, I really miss your point, and still believe the intro was better before your latest fixes. Paolo.dL (talk) 14:01, 18 December 2010 (UTC)

Did you mean perhaps that "eigenvector of a matrix" is formally incorrect? If this is what you meant, I disagree. I have never heard an expression such as "eigenvector of the multiplication by matrix A", or "eigenvector of the linear transformation performed by means of A" (Sorry, I really cannot understand your point, and this is what I deduce from your statements). Paolo.dL (talk) 14:10, 18 December 2010 (UTC)

I will try to be more clear. Space is not endowed with a "natural" coordinate system. Rotation of 90 degrees about "the" x-axis in given by one matrix in one coordinate system, but in a different coordinate system, the same motion in three dimensional space is given by a different matrix. We don't always use the same matrix.

To your second paragraph, no, I agree with you that "eigenvector of a matrix" is correct. I also never heard either of the expressions you quote. I'll go look at the article and, if either expression is there, I'll change it.

I want the article to avoid given false impressions, so if you point out particular sentences that are problematical, I'll try to improve them, or you can improve them.

Rick Norwood (talk) 14:51, 18 December 2010 (UTC)

I just checked. Neither of the phrases in quotation marks in is in the article. Now I'm confused. But I will try to move the more technical material to further down in the overview. Rick Norwood (talk) 14:56, 18 December 2010 (UTC)

I understand now what you meant about matrices in different bases. But I believe it is irrelevant to this discussion.

The intro before your recent edits stated that "A is a matrix". Your text states that "A is a linear transformation"... since we all know that A is most commonly a matrix (and Ax is a matrix mult.), you are actually saying "that the matrix "is" the linear transformation". Can you see how little your edits make sense to me? And how little your comment above explains them to me? (actually, your comment seems to be against your edits) Paolo.dL (talk) 16:05, 18 December 2010 (UTC)

Generalizations in the overview

The information you inserted in the "Overview" is correct and useful, but in my opinion was inserted too early. The overview was clearly written for somebody which knows little of scalar, vectors and matrices. This is why we shortly introduced these concepts. If right there, where the whole linear algebra is introduced in a few words, you want to generalize, that decreases readability from excellent to insufficient, and brings the article further away from the Featured article status. Trying to say everything immediately is the worst mistake you can do when you want to teach math or physics.

Generalizations should be introduced at the end of the overview. Indeed, there was a fascinating, intuitively appealing sentence about the generalizations, which you deleted completely, and replaced with your own text, which contains different information and in my opinion does not arouse the curiosity of the reader as well as the previous text did. Why??? Paolo.dL (talk) 14:49, 18 December 2010 (UTC)

As for "why", my effort is to be both clear and correct. As you know, that's not easy. I'll try again. Rick Norwood (talk) 14:53, 18 December 2010 (UTC)

Of course. But I can't understand why you did not like this text, which you rewrote from scrach, while in my opinion is well written, easily understandable, and intuitively appealing:

"Many kinds of mathematical objects can be treated as vectors: ordered pairs, functions, harmonic modes, quantum states, and frequencies are examples. In these cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenface, eigenstate, and eigenfrequency."

Paolo.dL (talk) 14:58, 18 December 2010 (UTC)

I do like that text. I took it out in a perhaps excessive attempt to avoid complexity. Feel free to put it back if you like it. Rick Norwood (talk) 15:06, 18 December 2010 (UTC)

I have no time to fix your edits. And I don't want to revert them. Please, move all your generalizations to the end, and find a place to move back this text. Paolo.dL (talk) 15:51, 18 December 2010 (UTC)

Will do. Rick Norwood (talk) 16:16, 18 December 2010 (UTC)

Scope of the article: matrices, linear, or non-linear transformations

How this article went downhill

I am the one who edited this article at the time it got promoted as a featured article. I was quite disappointed by the article as it is now. Today's editors just remember their first lectures in linear algebra but don't seem to know that the concepts of eigenvalues and eigenvectors were originally (and are still used) in order to characterize transformations (even if those transformation are not linbear!) and not matrices. They seem to believe eigenvectors are properties of matrices. They forget that a vector is something utterly different of an 1D-array of numbers. Just as a matrix is completely different of a linear transformation. Matrices and 1D-arrays of numbers are just special representations of vectors and linear transformations in a particular basis. I don't need to know whether a transformation of space is linear or not, whether it has a matricial representation or not, to decide whether a vector is an eigenvector or not. The concept of eigenvector is experimental: If a signal (e.g. V_in(t)=A*cos(w*t)) is provided to an electrical device and if this signal comes out only scaled by a factor without distorsion (e.g. V_out(t)=B*cos(w*t)) it is an eigenvector (if you want eigensignal) of the device. That this kind of things usually happen for a set of w only has led Helmoltz to introduce the concept of eigenfrequency. This has almost nothing to do with math -and a posteriori with any represention of a transformation in a basis- but with physical common sense! Vb (talk) 08:42, 22 December 2010 (UTC)

The advantage of the definition I propose for an eigenvector T(v)=λv is that it is the broadest definition of it and requires nothing to be explained except a series of examples for vectors and transformations which are very general species which require neither a finite dimensional vector space nor a linear transformation T but simply a device (a music instrument, a transistor) with an input out of a vector space and an output into a vector space. The article as it is now should be IMHO merged with Eigendecomposition of a matrix because it has nothing to do with the very general concept of eigenvalue. Vb (talk) 11:26, 22 December 2010 (UTC)

I congratulate you on getting the article promoted to featured article status, and I hope you will continue to edit here. However, I respectfully disagree that eigenvectors and eigenvalues have almost nothing to do with math. The math is the common ground of all of the applications -- in mathematics, statistics, and in science. If we were to define an eigenvector as a signal that comes out scaled without distortion, where would that leave the rest of the applications?

We all agree that the definition of an eigenvector v with an eigenvalue λ releative to some transformation T is Tv = λv. The article says this, though in stating that the transformation must be linear it may go too far -- the non-linear case should certainly be included. Also, your excellent example with signals deserves to be included in the lede. But since most people who encounter the concept see it first in a course in linear algebra, I think the example with "arrows" also belongs.

Please help improve the article. We need not only mathematicians, but also scientists (and statisticians). Rick Norwood (talk) 13:36, 22 December 2010 (UTC)

Well I was joking a bit when telling this has nothing to do with maths. Vb (talk) 14:41, 22 December 2010 (UTC)

Downhill or uphill?

In my opinion most people need a definition for the eigenvalues and eigenvectors "of a matrix". The expression "eigenvalues and eigenvectors of a matrix" is extremely common in textbooks (even used for titles of chapters or articles; see list of references above). According to GOOGLE, there are about 300000 pages containing the expression "eigenvectors of a matrix", and about 35000 containing the expression "eigenvectors of a transformation". This is the reason why the introduction starts with this definition. Not because of ignorance, but because of a choice. Generalizations can be effectively described at the end of the introduction. This is an effective approach which achieves two goals: (1) the most common definition is given immediately. (2) The first sentences are easy to understand. Paolo.dL (talk) 14:57, 22 December 2010 (UTC)

You are right Paolo. That's why if you want to write an article about this you can write one with the correct title, i.e. "eigenvectors and eigenvalues of a matrix." The name eigenvalue is more general than this: it applies to eigenfunctions, eigenstates, eigenfrequencies, etc. The concept of eigenvalue cannot be simply boiled down to the concept of eigenvalue of a matrix just because there are much more textbooks and pedagogical webpages for graduate students which limit themselves to applications with matrices than books on advanced calculus including integral transformations. If the prefix "eigen" has emerged that's not because these concepts are present in almost all math textbooks on linear algebra but because of the works of Helmoltz, Hilbert, Dirac and many others whose intention was not at all to solve textbook exercises. The question is whether we want to write one more textbook for graduate students or define these concepts for the layman i.e. for someone who doesn't even know what a matrix is. You may say it is not possible. I believe it is. That was the goal of the article I wrote, got featured, and was published on the main page. I think for example that Mona Lisa's picture (which I made) can be understood by anybody even by someone who doesn't know what a matrix is. Vb (talk) 15:49, 22 December 2010 (UTC)

For reference, it looks like it was featured on 1 November 2005.

I am on the fence on this. On the one hand, I agree that the more-general case is important and I agree with the sentiment behind it having "nothing to do with math"; on the other hand, the bulk of people looking for this concept will be looking for it in the context of finite-dimensional linear algebra. —Ben FrantzDale (talk) 20:14, 22 December 2010 (UTC)

Vb, you probably did not notice that the dilemma about which we are discussing is just how to start the intro, not whether or not it should (immediately or at the end) also give the most general definition. You maintain that it should start with the general case, others maintain that it should start with the most common case. Nobody ever maintained that this article should not introduce the most general concept (specific subsections and separate main articles exist about generalizations such as eigenfunctions, etc.). So, let's not fight about things on which we already agree.

Also, let's not forget that to understand this article, the reader must at least know the concept of vector, which a layman is not supposed to know. And the only difference between your approach and the current approach is that you want to start with the concept of "transformation" (or function, if you like), while others prefer to start with the concept of matrix multiplication. Your approach may be somewhat easier to understand for the reader, but not because it is "simple". It may be slighlty less complex, pehaps. Moreover, your approach does not achieve goal 1 (see my comment above).

I would also ask you to change the title of this section, if you don't mind, as "How the article went downhill" assumes that the current article is worst than your version, which as far as I know lost its "Featured article" status because somebody did not believe it deserved it, not because of our recent edits. Together with your initial judgement about our ignorance, it is not a nice start. I suggest "How the introduction was modified: general vs specific definition". Thank you.

— Paolo.dL (talk) 23:29, 22 December 2010 (UTC)

From my own experiences with math GA/FA's, I strongly suggest working on the introduction of the article last. The introduction just has to summarize the article and if the article is OK, what's in the intro will be almost dictated by the article's content. (Up to questions of presenting things). I agree with Vb that the concept of eigenvectors is more general than just matrices and the article does currently not convey this information entirely appropriately.

Here is my suggestion for the article's structure. (the question of how to introduce the concept is treated similarly as in group (mathematics)):

• 1. Lead section (again, don't work on this right now for your sanity's sake, it is premature at this point)
• 2. Introduction and definition: 2.1. give an easy example of (what amounts to) a 2-by-2 real matrix and discuss its eigenvalues and -vectors. Gently, but briefly, introduce the notion of "vector space", "linear transformation" by highlighting the important features of the example. 2.2. Take the last bits of the previous subsection as a motivation to state the definition of eigenvalues / eigenvectors of a linear transformation on a vector space. (What is currently described as "alternative definition" should not be more than a one-sentence remark, I believe.) 2.3. Give a second example of the concept, this time with an infinite-dimensional space. (Maybe a the example d/dx (f) = λ f would be good). Don't go in the differences of finite-dimensional vs. infinite-dimensional cases at this point.
• 3. A (long) section Eigenvalues and -vectors of matrices. Briefly recall the correspondence of matrices and linear maps. 3.1. Characteristic polynomial, 3.2. eigenspaces, eigenbasis, eigendecomposition. Point out that eigenvalues are independent of the choice of basis / independent under conjugation of the matrix. 3.3. Relation of eigenvalues to trace, determinant etc. (i.e., what is currently the section "Properties of eigenvalues"). Discuss these concepts maybe in the elementary 2-by-2 matrix example of the introduction. 3.4. Examples (from the current section "Existence and multiplicity"). This subsection should not be as long as it is now, I believe. 3.5. A section which does not exist at all yet: calculation (note that the current section heading "Computation of eigenvalues and eigenvectors" is a misnomer, in numerics there are other methods for calculating them, if the matrices are big).
• 4. Infinite-dimensional case. Again, take up the example of the introduction. Now discuss the difference to finite-dim'l case.
• 5. Applications
• 6. Generalizations. Discuss the case of a non-linear transformation (as per Vb's comments). Briefly mention what is currently the section "Left and right eigenvectors".
• 7. History (this does not necessarily have to be at this place, might be moved elsewhere in the article. But the mathematical concepts used in this section should have been developed before in the article, so it cannot come too early, IMO).

Jakob.scholbach (talk) 00:31, 23 December 2010 (UTC)

P.S. I concur with Vb that the number of textbooks merely introducing eigenvalues of matrixes is of little relevance, same as the number of google hits. Jakob.scholbach (talk) 00:35, 23 December 2010 (UTC)

No we don't simply discuss the lead: we discuss the whole structure of the article and which reader it is addressed to. With respect to the structure, I almost agree with Jakob. I however believe the first examples should not be easy for the graduate student but for the layman. A 2x2 matrix is too complex. We need examples which can be told without math. The example I provided above with the electrical signal is one of this kind (explained without cosine - of course). We had provided several others in the example section at [3]. I also believe the history section should be put just after this because it can be read by the layman without understanding the details. Important is here to explain when the things were done by whom in which context. The mathematical definitions and details (including eigenvalues of example 2x2 matrices) should come next. Then the layman will stop. He will maybe look further into the section Applications but that's all. Between History and Applications we have all the place we need to tell the reader about technical aspects. Vb (talk) 10:46, 23 December 2010 (UTC) —Preceding unsigned comment added by 89.0.174.251 (talk)

I agree that the article overemphasizes the case of matrices. However, I strongly disagree (as Vb's original post seems to suggest) that the notion of "eigenvalue" should be extended naively to operators that are nonlinear. This seems to be an unprecedented generalization that I have never seen in any of the literature. The eigenfrequencies of Helmholtz (see Helmholtz equation) the result of a linear operator. The fact that (idealized) waves satisfy the superposition principle is indeed the reason that eigenvalues are so powerful for studying them. By contrast, eigenvalues play virtually no role in nonlinear problems in mechanics. Sławomir Biały (talk) 13:40, 23 December 2010 (UTC)

I agre with Slawomir. The non linear case is not a usefull generalization. But the fact that the operator is usually linear does not help the reader to understand what an eigenvalue is. The linearity aspect is very important for practical applications, for the theory etc. but it is useless for understanding the concept. When I have a black box device (which maps a vector space into itsself) i don't care whether it is linear or not to decide whether the output is just a multiple of the input. Usually such black box thing is linear in a certain domain (e.g. frequency-domain) and is not in another one. So though we must emphasize that the concept of eigenvalue belongs to the linear equation realm we should refrain from telling this the reader before the technical part of this article starts. Vb (talk) 89.0.159.104 (talk) 20:15, 23 December 2010 (UTC)

Linearity is simply part of the context in which eigenvalues are meaningful and useful. It is the task of the article to convey this appropriately, even to novices. Sławomir Biały (talk) 13:19, 26 December 2010 (UTC)

Creating a separate article with narrower scope, focused on matrices

Since when the article was demoted, it was totally transformed. We cannot ignore this. The editors who decided to highlight the specific case of matrices knew exactly what they were doing, as they started from a very general article. They represented a huge amount of readers that are most interested in the specific case. The abundance of literature about eigenvalues and eigenvectors of matrices proves that the specific article is highly necessary. An article with a broader scope may be too broad, and pedagogically less useful for most readers.

On the other hand, it is legitimate and important to write an article which defines the general concept, using the T(v) notation, rather than Av. And this article should be called "Eigenvalues and eigenvectors". But the text of the current article, (except for the sections about generalizations such as "eigenfunctions") should be moved to a new article called "Eigenvalues and eigenvectors of a matrix". After broadening the scope of "Eigenvalues and eigenvectors", a warning should be provided at the beginning of it, stating that a general definition is given here, while those who are only interested in matrices should read [and edit] the specific article.

If we fail to understand the importance that a specific article has for the readers and for most editors, the general article will not last long, as too many editors will feel the need to emphasize the specific case, to provide what most people needs.

— Paolo.dL (talk) 18:52, 26 December 2010 (UTC)

Currently I don't see the necessity of a separate article. On the contrary, splitting off the more elementary stuff creates kind of a hole in this article. Also, I think the slightly more advanced linear algebra stuff like geoemtric multiplicities and characteristic polynomial is also not so much easier than what is currently in the infinite-dimensional section. The total length of the article is long, but not too long. Trimming down the very long and overly detailed section (here we are really crossing the lines of WP:NOTTEXTBOOK) "Existence and multiplicity" by at least 50% would give enough space to add short and gentle explanations of the basic linear algebra terms. (For example, I could think of a table summing up the examples very nicely, compare e.g. to the list here.) I think this can work perfectly well. After all a linear map is nothing terribly complicated. Jakob.scholbach (talk) 21:24, 26 December 2010 (UTC)

Jakob's suggestion seems like a sensible one. Of all the sections, "Existence and multiplicity" gets really bogged down in topics that are less relevant for a general purpose article. Perhaps some of this could be moved over to a different article like eigendecomposition of a matrix (which is in need of some gentler material)? However, overall I find the dual focus of the present article on matrices and linear operators to be quite appropriate, and I agree with the overall sentiment in this thread that a split is undesirable. Sławomir Biały (talk) 15:21, 27 December 2010 (UTC)

I don't see the point. Yes, the eigenvalues and eigenvectors of matrices are more specific than eigenvalues and eigenvalues of linear transformations in general. But they don't easily separate. First the matrix formulation is how most people learn about the topic, so excising matrices from this article will make it confusing to those unfamiliar with the topic. Second calculating the characteristic polynomial from the determinant of (A - λI) is the usual way to find the eigenvalues and eigenvectors, so can hardly be omitted. Every linear map can be written as a matrix, so though linear transformations are more general the matrix representation can always be used. And so T(v) and Av are just two different ways of writing the same thing. And yes, though it is a bit long that is best fixed by more judicious editing of some of the sections.--JohnBlackburne^words_deeds 21:44, 26 December 2010 (UTC)

I oppose splitting the article. Splitting the article would not solve the problem, because in the article on a general eigenvector the lede would still need to explain the concept to a layperson who didn't know what a vector was, which would put the elementry stuff right back in. Not everybody knows what a vector is, but everybody knows what an arrow is. Rick Norwood (talk) 13:29, 27 December 2010 (UTC)

Either double focus with emphasis on matrix formulation, or two separate articles

JohnBlackburne, you do see my point, as it is perfectly summarized by your sentence: "the matrix formulation is how most people learn about the topic". And I also perfectly agree with Sławomir Biały, when he writes: "I find the dual focus of the present article on matrices and linear operators to be quite appropriate". But since most contributors in this discussion seemed to think, as suggested by Vb, that the article should give a more general definition at the beginning of the intro and in the "Definition" section, and that the current article "overemphasizes" the matrix formulation, I was forced to try and convince you to create two separate articles:

1. Eigenvalues and eigenvectors. A more advanced and broader-scope article based on the general definition T(v) = λv, with the matrix formulation initially used just as an example, and with a warning at the beginning about the existance of a separate article focused on matrices. The matrix formulation, in this article, would be described after the general definition.
2. Eigenvalues and eigenvectors of a matrix. A shorter, simpler, less advanced, and narrower-scope article, for beginners, similar to many others in the literature, and focused on matrices (Av = λv). Much shorter and simpler, because it would not deal with generalizations. It would only contain a short sentence at the end of the intro, stating that the concept can be generalized, as described in a separate article.

The second article would be both extremely useful and perfectly sufficient for the huge amount of readers who (unfortunately) do not share your passion for advanced mathematics, do not give a damn about generalizations, and only need basic information about the most frequent application.

• Is this point so difficult to understand? I don't think so.
• Is my suggestion difficult to implement? I absolutely don't think so: article 1 is similar to the featured version of this article, while article 2 is similar to most articles and textbook chapters in the literature.
• Do I really want two articles? No, my main point is another.

I just wanted to remind you how little most readers know, how little would suffice to mix them up, and how consistent is the literature about teaching the topic to beginners using matrix notation. Perhaps, passion and deep knowledge (both of which I greatly respect) make some of you not too eager to grasp my point. This does not decrease my esteem for you all, but makes my task more difficult.

— Paolo.dL (talk) 11:48, 28 December 2010 (UTC)

I don't understand what you are suggesting. You earlier wrote "the text of the current article, (except for the sections about generalizations such as "eigenfunctions") should be moved to a new article", but now write "Do I really want two articles? No". Please make your mind up, or at least make it clear what you want, perhaps restating it with brief reasons to clear up any confusion.--JohnBlackburne^words_deeds 17:20, 28 December 2010 (UTC)

What I meant is explained right above, even in the title of this subsection. In short, the double focus in the current article, with emphasis on matrix formulation, is ok in my opinion. I do not agree at all that the "article went downhill", as stated by Vb. Since most contributors seemed to think the opposite, then I proposed to create a separate article focused on matrices. But my primary point was to keep the structure of the article as it is, and in this case, there would not be any reason to create a separate article.

For instance, I like the structure of the "Overview" section, which focuses on matrices and was not contained in the Featured version of this article. Rick Norwood and I (see our discussion above) agreed that the overview should maintain this structure, in which the first paragraphs focus on matrices (the most common application), and generalizations are briefly and brilliantly described only in the last paragraph(s). By the way, in my opinion the introduction should have the same structure as the overview. Thus, the sentence

• "... if A is a linear transformation... Av = λv".

should be

• "... if A is a matrix... Av = λv". (followed by short sentence about other kinds of linear transformations)

And even if you wanted to generalize it (too early, in my opinion) to linear transformations, it should be

• "... if T is a linear transformation... T(v) = λv".)

— Paolo.dL (talk) 21:34, 28 December 2010 (UTC)

Dimension

I've done a slight rewrite of some of the changes that were made yesterday. In particular, since we give the matrix definition earlier, I think that in the section titled "mathematical definition" we need to start with the more general case.

Near the end of this section is a short paragraph on "dimension". Do we need this here?

Rick Norwood (talk) 13:26, 29 December 2010 (UTC)

Here is my suggestion how to write the "Definition" section (if this is done properly, the current "Overview" section can be disposed of [i.e., some of the material there should go into the "Definition" section])

• As an example, give the linear map T(a, b) = (3a+4b, a-b) [or some other coefficients; we should have algebraic = geo multiplicity = 1 for both eigenvalues, probably good to have one positive and one negative eigenvalue; the entries of the matrix should be pairwise different integers] from R^2 to R^2. We don't call it linear map at this point (in order not to scare people off). Illustrate the mapping with a picture, pointing out two eigenvectors and -values (also in the image). Mona Lisa is nice, but it is (IMO) difficult to actually see the eigenvectors. I personally prefer something like this. Introduce the concept of matrix by calling a matrix a handy way of summarizing maps as the above. Recall/briefly explain that T(a, b) = Ax, where x=(a, b). Then, define eigenv's for matrices.
• (Possibly in a new subsection:) point out that R^2 is an example of a vector space and that T is an example of a linear map. (Find a wording which encourages the reader to jump to the examples etc. if (s)he does not want to go into abstract algbra.) Then, repeat the definition for a linear maps between vector spaces. Point out that the previous definition is a particular case of this. (Do not go into how matrices and linear maps are related. This should be done later, when the independence of eigenv.'s of matrices under conjugation (i.e., choice of bases) is discussed).
• We might conclude this section with an infinite-dimensional example, or we might leave this for the section containing the spectral theorem etc. Right now I feel it might be better to postpone it.

What do you guys think about this plan? Jakob.scholbach (talk) 17:17, 29 December 2010 (UTC)

@Rick: what paragraph do you mean? I would remove the one "Every vector space has a fixed dimension...." from the Definitions section. We don't need to know/tell what the dimension is at this point. We simply say: n-by-n matrix and vector space (of an unspecified dimension). The first point where we need the notion of dimension is at geometric multiplicities, I believe. Jakob.scholbach (talk) 17:17, 29 December 2010 (UTC)

I think we need a clear transition between the introductory matrix section and the "Mathematical definition" section, rather than disposing of the overview. We certainly need at least one infinite dimensional example -- probably the derivative -- but not until later in the article. I suggest postponing the material about dimension and n by n until later also. Rick Norwood (talk) 19:59, 29 December 2010 (UTC)

Table

With my default settings and browser width of about 1000 pixels the following table is too wide: I can only see three of the four columns without scrolling. This is caused by the images: the three thumbnails and the LaTeX formulae rendered as PNGs which stop it resizing to fit. I think the same can be achieved, i.e. surplus calculations can be removed, without a table, especially as the descriptions are still needed. Example 1 and Example 2 could also be merged - there are still too many 2D examples compared to 3 or higher dimensional examples.--JohnBlackburne^words_deeds 23:24, 29 December 2010 (UTC)

	Horizontal shear	Scaling	Unequal scaling	Counterclockwise rotation by $\varphi$
Illustration	Horizontal shear. The shear angle φ is given by k = cot φ, where k is the shear factor	When a surface is stretching equally in all directions (a homothety) each of the radial vectors is an eigenvector.	Vertical shrink (k2 < 1) and horizontal stretch (k1 > 1) of a unit square.
Matrix	${\begin{bmatrix}1&k\\0&1\end{bmatrix}}$	${\begin{bmatrix}k&0\\0&k\end{bmatrix}}$	${\begin{bmatrix}k_{1}&0\\0&k_{2}\end{bmatrix}}$	${\begin{bmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{bmatrix}}$
Characteristic equation	$\lambda ^{2}-2\lambda +1=(1-\lambda )^{2}=0$	$\lambda ^{2}-2\lambda k+k^{2}=(\lambda -k)^{2}=0$		λ2 − 2λ cos φ + 1 = 0
Eigenvalues λi	λ1=1	λ1=k	λ1 = k1, λ2 = k2	λ1,2 = cos φ ± i sin φ = e ± iφ
algebraic and geometric multiplicities	n1 = 2, m1 = 1	n1 = 2, m1 = 2	n1 = m1 = 1, n2 = m2 = 1	n1 = m1 = 1, n2 = m2 = 1
Eigenvectors	$\mathbf {u} _{1}=(1,0)$	$\mathbf {u} _{1}=(1,0),\mathbf {u} _{2}=(0,1)$	$\mathbf {u} _{1}=(1,0),\mathbf {u} _{2}=(0,1)$	$\mathbf {u} _{1}={\begin{bmatrix}1\\-i\end{bmatrix}},\mathbf {u} _{2}={\begin{bmatrix}1\\i\end{bmatrix}}.$

OK, I see. Of course, feel free to improve the layout and everything else. I guess in this case we could move the image captions to the text, and the characteristic equations could also be formatted in a way that makes them less wide.

Generally speaking, my recent edits are just meant to reduce the enormous amounts of useless redundancy that was (and partly still is) in this article. I totally agree that Examples 1 and 2 should also be merged with that table. Maybe we can afford to have one "worked" example. Example 2, though, is just a particular case of the "Unequal scaling" example. About higher dimensions: sure, a 3D example would be nice. Who is up to create a nice pic for a "unequal scaling" map in 3D? Jakob.scholbach (talk) 23:40, 29 December 2010 (UTC)

P.S. To be safe: by "useless redundancy" I do not mean thorough explanations of the concepts, which in a few places are quite well-done, IMO. Rather, I'm talking about repeating material in various sections scattered all over the article, which makes it very difficult for readers to actually find the stuff.

In addition to this kind of redundancy, I feel we have to cut a little bit the textbookish writing style used in some places. For example, once the eigenvalues and -vectors are calculated, checking that they are really the right ones is like a textbook, something we should avoid. Jakob.scholbach (talk) 23:44, 29 December 2010 (UTC)

Horizontal shear

Horizontal shear. The shear angle φ is given by k = cot φ, where k is the shear factor
Matrix
${\begin{bmatrix}1&k\\0&1\end{bmatrix}}$
Characteristic equation
$\lambda ^{2}-2\lambda +1=(1-\lambda )^{2}=0$
Eigenvalues λi
λ1=1
algebraic and geometric multiplicities
n1 = 2, m1 = 1
eigenvectors
$\mathbf {u} _{1}=(1,0)$

I notice that the table is followed by textual descriptions of the various transformations. It seems to me that a better layout would be to take the individual columns of the table and put those into the relevant subsections like "infoboxes". For visually-oriented learners, it is better to have the image in the immediate vicinity of the text. See example on the right. Sławomir Biały (talk) 13:21, 5 January 2011 (UTC)

Hm. I agree that the table is not the optimal layout. However, your suggestion has a similar drawback: the individual tables are very long and if we put them right to the usual text, the four tables would be way longer than the text describing these tables (at least in my current window widths etc.) Maybe another idea? On the other hand, I feel at this point it is maybe too early to think about layout, since the actual content is still subject to change a lot. Jakob.scholbach (talk) 14:47, 5 January 2011 (UTC)

What is the "Overview" for?

I have some objections about the shape of the Overview section. Since some discussion above and thoughts etc. have been invested, it's probably more prudent to ask first before completely overhauling this section. Basically, my question is: why do we have such a separate section? Parts of this constitute a separate lead section (thus, if the lead would be well-written, create redundancy, which we should avoid). Other parts of this are relatively unspecific bits of linear algebra which we either don't need at all or, I believe, need in a way that is more closely connected to the rest of the text. For example, describing a vector as an arrow is helpful, but a closer integration of this information to the Definitions section would simplify understanding both parts. Finally, other parts are already (and have been so before my recent edits) contained in the Definition section, again creating useless redundancy. Does anyone object to moving these parts where they (IMO, as per the above) belong? Jakob.scholbach (talk) 00:13, 2 January 2011 (UTC)

I agree. The lede should contain the informal definition and terms, the Definition should specify things more formally, between them they should cover all the relevant points. Other than that the Overview section contains some confusing linear algebra which readers can better find in that article.--JohnBlackburne^words_deeds 00:51, 2 January 2011 (UTC)

I also agree. I think the next to last paragraph in "overview" should be moved up into the lede, and all the other information in "overview" moved further down in the article if it is not there already.Rick Norwood (talk) 14:45, 2 January 2011 (UTC)

Rick's recent edits

I'm somewhat unhappy about parts of Rick's recent edits: IMO (and from the above discussions it seems that most? other editors share this opinion) it is undesirable to define eigenv's for linear transformation first. Quite a bulk of the readers of this article will not /need not know linear algebra, hence won't know vector spaces and linear maps. We may regret this or not, but I think we definitely should offer a separate, first definition of eigenv's for matrices. We can easily afford that, since it takes little space. Much of the intuitive picture of the vector changing direction will not be understandable in the abstract context and most of the early examples are eigenv's of particular matrices anyway. So actually removing the matrix-eigenv-definition is not only creating a leap for the reader to overcome, but also leaves an incoherent article structure. What do you, Rick Norwood and others, think? Jakob.scholbach (talk) 19:55, 3 January 2011 (UTC)

P.S. IMO, removing redundancy, as I recently did, by removing duplicate information is an improvement, but removing small steps in an explanation of a more abstract/advanced etc. piece of information is not a priori helpful. Jakob.scholbach (talk) 19:55, 3 January 2011 (UTC)

The matrix definition now appears twice, once in the lede and once in the definition section, and the general definition is now at the bottom of the definition section. Let me know what you think. Also, I'd appreciate input on the "small steps" you would like restored. Rick Norwood (talk) 13:27, 4 January 2011 (UTC)

The Definitions section starts out mentioning linear transformations and talks about "establishing Cartesian coordinates". I think neither is necessary nor helpful for the unoriented reader. Let's simply stick to a matrix is an array of numbers and an Euclidean vector is a sequence of numbers. Do you agree?

By "small steps" I meant starting out with these very basics and then, somehow gradually, get to the more abstract parts.

Also, but this is probably just something you oversaw?, at the end of that section, the definition of eigenv's for linear transformations is repeated twice. Jakob.scholbach (talk) 14:54, 5 January 2011 (UTC)

Here is why I have a problem with the idea that a vector "is" a sequence of numbers. Many students get the idea that a vector "is" a sequence of numbers, and when that idea gets stuck in their heads, they have a lot of trouble getting over it. It is easier to teach something correctly in the first place than to teach it incorrectly and then require students to unlearn something they've learned. I want the lede to be simple, but not wrong. A vector "is" a sequence of numbers relative to some fixed basis. Relative to a different basis, a different sequence of numbers represents that vector. I'm sure you know this, but students have a lot of trouble with the concept.

I'll take a look at the repetition you mention.

Rick Norwood (talk) 20:32, 5 January 2011 (UTC)

Insisting on a coordinate-free approach to eigenvectors is pedagogically misguided in my opinion, and at any rate most textbooks in linear algebra start with defining eigenvectors in Euclidean space. I think there is a large consensus among math editors at wiki that Bourbaki style of going from general to particular should be avoided in wiki. Tkuvho (talk) 20:59, 5 January 2011 (UTC)

I think it is fair to say that an Euclidean vector is a sequence of numbers. Much the same way as a linear map is a matrix up to choices of bases, a vector (in an abstract vector space) is a sequence of numbers, again up to choices of bases. Therefore, I think we can safely describe the matter this way. Agreed? If we agree about this (which I hope), it is not necessary to mention the abstract notions right at the beginning. On the contrary it will scare away a portion of our readers. I would suggest discussing the relation of eigenv's of a matrix vs. a linear transformation at the spot where we mention that eigenv's are independent under conjugation of the matrix (surprisingly, this bit is not yet in the article!). Jakob.scholbach (talk) 01:55, 6 January 2011 (UTC)

By the standards of linear algebra textbooks, the use of the term "sequence" in this context is unusual. It is much more common to speak of n-tuples. Of course we should start with eigenvectors as n-tuples. Tkuvho (talk) 06:11, 6 January 2011 (UTC)

Please stop joking. A Euclidean vector is an arrow which define a translation in a Euclidean space. Euclid is a bit older than Descartes. I believe. A Euclidean space is a space where the Euclidean geometry can be applied not the Cartesian space Rⁿ. Vb (talk) 17:07, 9 February 2011 (UTC)

Cartesian coordinates.

I certainly am not trying to write like Bourbaki! The parts of the article I have written are as concrete as I can make them and still be mathematically correct.

Is this the offending sentence? "Once a set of Cartesian coordinates are established, a vector can be described relative to that set of coordinates by a sequence of numbers." It seems to me that anyone coming to this article has learned Cartesian coordinates in secondary school. After that one sentence, the article passes immediately to the idea of n-tuples.

In any case, this article is not where vectors are defined -- and the article Euclidean vector does not begin with n-tuples. Most of the books I'm familiar with begin with the idea that a vector has magnitude and direction, and only then go on to the idea of an n-tuple, usually after introducing the vectors i, j, and k -- that is, after introducing Cartesian coordinates.

Rick Norwood (talk) 13:01, 6 January 2011 (UTC)

I don't know if the sentence is "offending" anyone or not, but I think it is pedagogically unsound. Both of us feel that the idea of a vector space is a natural one, but if you try to remember how you learned about vector spaces you will discover that you learned lots of examples first. To someone who is not familiar with this notion, the sentence is meaningless, not offending. Tkuvho (talk) 13:32, 6 January 2011 (UTC)

I actually came to the talk page to ask a question about this particular phrasing and stumbled onto the section. As a "I've taken my calculus branch in college student", it immediately struck me that "Cartesian coordinates" could be any coordinate system and not simply the Cartesian. Is that supposition true, and if so, maybe it should be changed/tweeked? I suspect it's supposed to be consistent with the example of the vector in n-d space, but then it should be written as an example. --Izno (talk) 01:16, 30 May 2011 (UTC)

Mona Lisa

It seems to me (and to Rick Norwood, above) that the Mona Lisa example appears to be rotating in three-space rather than shearing in two-space.

Version in the article

Does it help to add axes to the image, like this?

with axes added

To me, sadly, the axes don't seem to help. Any other ideas? I really like the basic notion of using the Mona Lisa to demonstrate the non-movement of the eigenvector in shear.

(cross-eyed) TreyGreer62 18:22, 21 January 2011 (UTC)

How about like this: skewed the other way. The one that's there looks like the left edge is further back in part as it's darker. That it seems to face inward, like a book or half a panel, might also play a part. skew it the other way and the effect is a lot less – at least to me it looks a lot less 3D. --JohnBlackburne^words_deeds 18:54, 21 January 2011 (UTC)

skewed the other way

Wow John, that's pretty amazing. Your version looks much less 3D to me as well. You inspired me to try again, this time with the eigenvector on the horizontal axis and with an overlaid grid. Cropping to the rectangle of the unskewed image seems to help as well. TreyGreer62 (talk) 20:45, 21 January 2011 (UTC)

Horizontal skew, with grid, cropped to rectangle.

I'd vote for yours too: the grid lines make it clearer, the arrows are neater and skewing it that way and cropping it there's no way it looks like it's rotated in 3D. She's less pretty with lines all over her but we're doing math not art !--JohnBlackburne^words_deeds 21:10, 21 January 2011 (UTC)

I agree: The new pic with the grid is much better than mine! Vb (talk) 13:38, 2 February 2011 (UTC)

It looks pretty nice. We should not have bright red and bright green for accessibility to colorblind people. —Ben FrantzDale (talk) 12:22, 3 February 2011 (UTC)

I've replaced the image with the latest one, fixing the arrow colour as suggested.--JohnBlackburne^words_deeds 00:22, 14 February 2011 (UTC)

Great! Rick Norwood (talk) 13:11, 15 February 2011 (UTC)

Eigenvalue > 1

For those of us struggling to understand this, it would be a good idea to have a simple example where the eigenvalue is greater than 1. The example for 1 is OK, but what about say 5? Myrvin (talk) 19:52, 7 April 2011 (UTC)

please motivate in non-mathematical terms

Similar to most "mathematical" wiki pages, this page (IMHO) suffers from the common malady: "if you don't already know the math, the page is useless." I would like to see an introduction, or preliminary section, that explains what eigenvalues/eigenvectors are, and why they are useful, with absolutely no reference to mathematics.

I realize this is *extremely* difficult. — Preceding unsigned comment added by 108.77.141.166 (talk) 03:25, 6 July 2011 (UTC)

I, and others, have tried to do this, but it always gets reverted. I would like to see something like this in the lead.

An eigenvector of a transformation of space, such as a rotation or stretching, is a vector that still lies in the same line after the transformation. If space is rotated about an axis, an arrow pointing along the axis of rotation is an eigenvector. If space is stretched along a north/south axis, an arrow pointing due north or south is an eigenvector. The eigenvalue of an eigenvector is the amount of stretching.

Comments? Rick Norwood (talk) 14:12, 6 July 2011 (UTC)

I'd like to see some of that too. One suggestion I have would be to replace the much-discussed Mona Lisa example with one using not a shear but a simple off-axis squishing. The problem is that the shear has complex eigenvalues which gets nasty and is a turn-off. In a great number of real-world domains, someone can get away with understanding eigenvalues and eigenvectors only in the context of symmetric matricies/operators, where all the eigenvectors are nicely orthogonal and all the values are real. It is much easier to explain that special case without getting mathey before diving into the complex cases. —Ben FrantzDale (talk) 15:42, 6 July 2011 (UTC)

Have a look at the version of this article when it used to be featured! Here! The 2005 editors paid much attention to this aspect! — Preceding unsigned comment added by 89.0.188.47 (talk) 14:15, 15 March 2012 (UTC)

"eigen" - Dutch or German?

The Dutch article says that this terminology was introduced in 1904 by the German mathematician David Hilbert (1862-1943).

217.235.233.145 (talk) 11:13, 22 September 2011 (UTC)

And wiktionary agrees: wikt:eigen-. Reverted it.--JohnBlackburne^words_deeds 11:45, 22 September 2011 (UTC)

Change "Tx" to "T(x)" in Formal Definition

Every transformation is a function, so isn't T(x) more correct?

I'm just a student, so I'm not changing this myself. --Nyellin (talk) 14:18, 18 February 2012 (UTC)

Definition - multiplication commutativity

Throughout the entire definition section, it is stated several times:

... the multiplication of a vector x by a square matrix A ...

Multiplication of matrices is not commutative, so x*A does not make sense. Should be rephrased everywhere to state A*x — Preceding unsigned comment added by Previously Used Coffins (talk • contribs) 13:40, 18 April 2012 (UTC)

• Fixed in the introduction (at least) by distinguishing right and left eigenvectors. BTW: the body states that left and right eigenvectors "are the same for Hermitian matrices". Should it be "symmetric" instead (even in the complex case)? The defining equation is vA = λv not v^*A = λv, correct? --Jorge Stolfi (talk) 04:17, 8 August 2012 (UTC)

Is Animation Misleading

Am I missing something or is the animation misleading. While the Blue lines with arrows in the annimation (also the purple lines with arrows) representing how the distance vector between two points increases with every itteration of the transformation remains parallel to the eigenvector, they are not in themselves the vectors that are been transformed. Each of these vectors are a result of the subtraction of the vectors representing the two points been transformed, and the magnitude and direction of these vectors do indeed change once transformed.

Only the vectors on the line at 45 degrees to the origin as indicated by the blue points don't change direction. — Preceding unsigned comment added by 109.76.126.55 (talk) 13:45, 14 June 2012 (UTC)

Vectors aren't stuck to the origin, they can be anywhere. They only have magnitude and direction, not position. The scattered arrows represent multiples instances of several vectors and how the transformation operates on the vectors across the space where the transformation was performed, as they would be found as the relative position of points in space. Showing only vectors at the origin, or the eigenvector space as two lines, would miss the purpose of the animation, IMO. But I can understand why it could be considered misleading. Maybe I'm in the wrong and it is. Well, whatever. I'll just remove it. If anyone feels like adding it back, feel free to do so. If it's considered bad, I'll gladly ask for a mod to speedy delete it. — Kieff | Talk 03:09, 28 June 2012 (UTC)

It seems fine to me. They're not position vectors (so starting at the origin) but displacement vectors (so starting anywhere), but they behave as eigenvectors: they remain parallel (at least the blue and magenta ones do). It's a better way to show this behaviour for multiple vectors as if you put all the vectors at the origin they'd overlap and obscure each other. I'll add it back as you also seem to agree with my interpretation of it, and the IP's assertion that only the vectors on the 45° line don't change direction is simply wrong.--JohnBlackburne^words_deeds 03:35, 28 June 2012 (UTC)

Real eigenvalues

Currently the gallery of examples includes a rotation matrix which has complex eigenvalues. Students find in the Jordan normal form article that such complex eigenvalues are sometimes unavoidable in real matrices. To improve the accessiblility of the article, the rotation matrix might be replaced by

$A={\begin{pmatrix}\cosh(\phi )&\sinh(\phi )\\\sinh(\phi )&\cosh(\phi )\end{pmatrix}}.$

As shown by an example in the characteristic polynomial article, this matrix has real eigenvalues that are reciprocal, hence it is similar to a squeeze mapping. While introduction of complex numbers is inevitable in algebra, doing so at this point in the development may be confusing.Rgdboer (talk) 21:59, 31 August 2012 (UTC)

This important article has 69,000 readers per month and 288 watchers. In one month some response was expected. Hearing the silence, changes were made today to improve the article by including the above matrix that has real eigenvalues in the gallery. In addition, a section was included on "complex eigenvalues" since the topic arises with rotation. The fact that a matrix may fail to have an eigenvalue over the real field might be mentioned in the lede.Rgdboer (talk) 21:59, 28 September 2012 (UTC)

Extended animation

Eigenvectors animation, extended.

By a couple of requests, I just created an extended version of the animation, showing all quadrants and both eigenvector axes. I'm not sure if it's necessary to be included in the article, as it requires more space for a good visibility and the original seems fine enough, at least to me. Either way, just leaving it on record for anyone who may think it's useful. — Kieff | Talk 03:23, 12 October 2012 (UTC)

Proposed move to Eigensystem

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was not moved. --BDD (talk) 17:39, 19 October 2012 (UTC) (non-admin closure)

Eigenvalues and eigenvectors → Eigensystem – Per WP:AND, [w]here possible, use a title covering all cases. 220.246.154.177 (talk) 23:14, 12 October 2012 (UTC)

• oppose. Eigensystem is a far less natural and familiar name. I mean the way I learned about them was as eigenvectors and eigenvalues: both concepts were introduced at the same time but they were called by those names. The system or systems were vector and matrix algebra. Eigenvalues and eigenvectors were just properties of them, like determinant, trace, inverse, etc.. See also the external links which almost all use these names.--JohnBlackburne^words_deeds 23:35, 12 October 2012 (UTC)
• Oppose, for the same reasons JohnBlackburne mentioned. The terms "eigenvectors" and "eigenvalues" are universally accepted and within the scope of the current article, and these are used throughout the literature as well. It's better to stick with it. — Kieff | Talk 02:26, 13 October 2012 (UTC)

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

How to find Eigenvalues: Main example is misleading

So when describing initially how to calculate the eigenvalues of a matrix, the article gives an example problem. It's very misleading, however, since the example is of a diagonal matrix, which may have an elegant solution, but implies a purpose for its diagonal-ness beyond pure elegance (if you didn't happen to remember the preceding lines of context perfectly). For those trying to relearn the topic, like myself, it'd be greatly appreciated if someone were to remove the unnecessary context (the use of the diagonal matrix), especially since the whole concept is difficult enough as it is, and simply replace it with a 3x3 or 4x4 matrix full of random values. I'd edit it myself, except I'm not entirely brushed up on the subject...

Thank you! --Frizzil (talk) 07:16, 22 October 2012 (UTC)

I agree that a diagonal matrix is misleadingly simple, but the simple examples shown are actually symmetric matrices, which is a very common case for real-world eigenvalue problems. Using a matrix of random values will almost certainly give you complex eigenvalues which I would argue is overly general for an introductory example. Eigendecomposition of real symmetric matrices has a pretty simple geometric interpretation. Also, showing that a diagonal matrix has its eigenvalues on its diagonal -- that is already eigendecomposed -- is a critically important concept. Do you have a real-world example of eigendecomposing a non-symmetric matrix? —Ben FrantzDale (talk) 11:37, 24 October 2012 (UTC)

Unfortunately I don't-- again, I'm just brushing up on the subject, as I only learned it at the very end of my course on matrix theory. I apologize for any ignorance on the subject, I just thought it could be simplified by, well, unsimplifying it. Maybe it could just be rephrased to make the diagonal-ness more obvious? I'll take a look at it. --Frizzil (talk) 21:47, 11 November 2012 (UTC)

I rephrased the text of the example to clarify it. Perhaps the entire example needs attention from an expert in math formatting? --Frizzil (talk) 22:02, 11 November 2012 (UTC)

Applications

One of the main applications of eigenvalues/eigenvectors is in the solution of linear ODEs. The applications section has a section on vibrations, but there should be a more general section on solutions to ODEs and how/why they are related to the characteristic equations of linear ODEs.

Thanks. 128.227.239.245 (talk) 15:01, 24 January 2013 (UTC)

Sections that should be moved elsewhere

The following sections should not be in this article and should be moved to (merged into) more specialized articles, leaving here only a short sentence mentioning their existence:

• Eigenvalues and eigenvectors#Spectral theory
• Eigenvalues and eigenvectors#Eigenfunctions
• Eigenvalues and eigenvectors#Associative algebras and representation theory

Also, the table of geometric transformations seems useful, but the following subsection (that describes the geometric effect of the transformations) is largely out of topic and should be moved to some article related to analytic geometry:

• Eigenvalues and eigenvectors#Geometric description

The following subsections of "Applications" are too confusing to help. They should be rewritten for general readers, or moved to specialized articles with only a brief mention here:

• Eigenvalues and eigenvectors#Schrödinger equation
• Eigenvalues and eigenvectors#Molecular orbitals
• Eigenvalues and eigenvectors#Geology and glaciology
• Eigenvalues and eigenvectors#Example: waves on a string

The last one is misplaced (should be in Applications) and fails to explain why eigenfunctions are relevant to the problem.
All the best, --Jorge Stolfi (talk) 02:29, 4 February 2013 (UTC)

General math cleanup

Hi, I just did a massive edit with the following changes:

• Converted all the pseudo-math formulas (with wiki-italics, Greek unicodes, "math" and "mvar" templates) to the <math>...</math> notation. In my browser (Chrome) these now display quite effectively and neatly, apart from a few line breaking and spacing bugs. If this is not the case for everybody, please report here and I will consider changing some of them back.
• Removed some unnecessary boldface. (The use of boldface for vectors, traditional in some engineering fields, has its merits in books or specialized papers; but is generally useless or worse for Wikipedia articles, considering that the target public can be assumed to have a high school or college background and are therefore unfamiliar with the convention.)
• Elaborated some of the basic examples, such as characteristic polynomials.
• Changed the notation for algebraic and geometric multiplicities to indicate that they are properties of the eigenvalue and of the operators. Moved the geometric multiplicities up, to the section where eigenspaces are discussed.
• Added a rotation transform to the table of geometric transforms.
• Removed almost all no-breaking spaces. (Cosmetic edits like preventing bad line breaks should be a very low priority goal, since readers come here for information not beauty. No-reaking spaces in particular make the wikisource much harder to edit, and this more than negates their positive value. Besides, most line breaking problems should be fixed invisibly -- in the browser, in the server, and/or in the javascript -- and not by the editors in the wikisource.)
• Removed some duplication in the text.
• Tried to clarify some parts, such as
  • The "bi-infinite shift" example for Spectral Theory section.
  • The diagonalization and eigendeomposition of a matrix
  • That non-real complex eigenvalues of a real matrix come in pairs
  • That left eigenvectors are right eigenvectors of the transpose.
  • That once aneigenvalue is known, the eigenvectors can be found by solving a linear system.

Hope I did not add too many errors. All the best, --Jorge Stolfi (talk) 02:16, 4 February 2013 (UTC)

I had to revert the changes as I encountered a lot of markup errors. The TeX renderer gave me dozens of errors and there were several

       preformatted text blocks like this

all over the place. I don't agree that expressions such as "3 x 3" or the name of a matrix such as simply "A" should be inside <math> tags, but if you do decide to use those, then be absolutely sure the contents of the tag are valid LaTeX code! They should also have consistent typesetting, such as the matrix name being a bold A.

For one, in my user preferences, I set it to always render math tags as PNG images. This is why I got to see all the errors and you probably did not. I advise you to temporarily change that setting to always render PNG just to be sure everything is working properly. And of course, double-check everything with the preview function before committing the changes.

I haven't had the time to look through all the other edits you've made to the rest of the page yet, but I just wanted to point out that urgent issue. — Kieff | Talk 02:24, 4 February 2013 (UTC)

• Indeed there were a couple of things that MathJAX understands but the Wikipedia math-to-PNG renderer does not: "×" instead of \times, \phantom, "…", and some other UNICODE weirdos that I was unable to see but went away once I re-typed the offending formula. I have changed my Wikipedia profile to use the PNG renderer and now the page displays without errors.
  I agree that the PNG option makes the page rather too heavy; it also renders things in the wrong font size. Is it too unrealistic to expect people to switch to MathJAX yet? It seems to be the way of the future...
  As for using boldface for matrices and vectors, see the comment above. It not a general convention (mathematicians and physicists do not seem to use it), it is not needed in a text of this size, and it makes editing more painful. I believe the page is now consistent, without bold for matrices and vectors.
  All the best, --Jorge Stolfi (talk) 03:45, 4 February 2013 (UTC)

---

I do not understand a maniacal addiction of certain users to <math>. With MathJax, versions of Jorge Stolfi requires about 20 seconds to be rendered in my browser. I think that with PNGs it easily can cause a vomit. Why he uses this resource-consuming <math> to say “$a$”, “$\mathbf {v}$”, and “$2\times 2$” while the same is available for a much lower cost? Incnis Mrsi (talk) 06:52, 4 February 2013 (UTC)

• See my reply below. Indeed I may have abused <math>, but I believe it is the way of the future. Efficiency problems will be solved. The way a looks on screen depends on the chosen "skin". Having two ways to enter formulas is very bad for editors (especially for novices) and even for looks. --Jorge Stolfi (talk) 15:59, 4 February 2013 (UTC)

  You can think it’s good, bad, or whatever you want, but three ways to enter formulas is a fact. I think the plurality is good. You think it’s bad, but there is no consensus in favour of the exclusive use of the current implementation of <math>. In any case, Wikipedia does not need a human job for conversion from simple {{math}}/{{mvar}} to <math>. A bot-like job could be performed with bots. You are human, so do fix formatting where it is really poor. Incnis Mrsi (talk) 17:20, 4 February 2013 (UTC)

Destruction of &nbsp;

With spaces	With  
If we think of a vector x as a single-column matrix with n rows, then the linear operator defined by a matrix A with n rows and columns maps the vector x to the matrix product Ax. If we think of a vector x as a single-column matrix with n rows, then the linear operator defined by a matrix A with n rows and columns maps the vector x to the matrix product Ax. If we think of a vector x as a single-column matrix with n rows, then the linear operator defined by a matrix A with n rows and columns maps the vector x to the matrix product Ax.	If we think of a vector x as a single-column matrix with n rows, then the linear operator defined by a matrix A with n rows and columns maps the vector x to the matrix product Ax. If we think of a vector x as a single-column matrix with n rows, then the linear operator defined by a matrix A with n rows and columns maps the vector x to the matrix product Ax. If we think of a vector x as a single-column matrix with n rows, then the linear operator defined by a matrix A with n rows and columns maps the vector x to the matrix product Ax.
Attention! Samples compare merits of &nbsp; vs U+0020, not {{math}} vs <math>!

After edits of Jorge
Stolfi all “matrix {{mvar|A}}” (and similar), which I put into the article, became “matrix <math>A</math>”. I already said about addiction to <math>, but what is a substantiation for destruction of &nbsp;? Incnis Mrsi (talk) 07:12, 4 February 2013 (UTC)

• Well, sorry for having erased the work you invested in adding the non-breaking spaces. I would gladly put them back if I could be convinced that they are a positive thing. Please read my arguments against their use.
  As for excessive use of <math>...</math>, you do have a point: by the same argument that contents and easy of editing are more important than looks, we should avoid it. And indeed I sould not have used math for "3×3"; and I will fix that. On the other hand, I believe that TeX/math should become the preferred way of entering formulas in any article with more than one or two formulas, and that the {{math}}/{{mvar}} should be avoided. Please read my arguments on this topic. <math> .
  Thus, I am sorry for all your work, but I would really object to adding back the non-breaking spaces and {{math}}/{{mvar}} templates. All the best, --Jorge Stolfi (talk) 15:50, 4 February 2013 (UTC)

  Are there, actually, arguments relevant to advanced editors? Say honestly: you just don't like all this clumsy formatting stuff. Let us compare readability of the text with &nbsp; and with ordinary spaces. Incnis Mrsi (talk) 17:20, 4 February 2013 (UTC)

  • On my browser (Chrome) and skin, I see absolutely no difference between the two columns. That creates a problem, right? How will an editor know whether to use NBSP or not? --Jorge Stolfi (talk) 18:38, 4 February 2013 (UTC)

    Try to alter style="width:…em" parameter in the table (simultaneously at both cells) and to push Preview. “Inconvenient” spaces can word wrap, but for an arbitrary text width there is a possibility that no “inconvenient” wrap occurred in a given text. Incnis Mrsi (talk) 19:00, 4 February 2013 (UTC)

    • I got to see some effect by squeezing the browser's window. An "inconvenient wrap" would be a break before a formula? If so, then editors would have to add an NBSP before every formula, since they cannot predict where the breaks would occur in other people's browsers. That is not acceptable. This is definitely a problem that should be fixed at the systems level, not by us editors. --Jorge Stolfi (talk) 19:13, 4 February 2013 (UTC)

      Before every formula? There is no "before" and I did not say anything about "every"; you can recall it in the version saved by me. I try to bind such non-verbal items as A and x with their heads (wherever it is possible; where it is not possible I do not). Although it may cause a controversy and eventually be rejected, you are not an owner or manager of this project where you could decide to "purge this nasty &nbsp;s now because they irritate me". I do not believe that my &nbsp;s actually hindered your edits of the wiki code to such extent that you opted to delete all of them only to alleviate your subsequent editing. Incnis Mrsi (talk) 21:21, 4 February 2013 (UTC)

• Sorry, I got it now. (The intent was not obvious, was it? I assumed they were attempts to control the spacing around formulas, which sometimes get mangled when using HTML math.)
  I have never seen technical writers attempting this sort of fine line-breaking control, not even for the most finnicky technical journals. Perhaps because TeX already does a very good job at line breaking, and avoids breaks before formulas if it can. (That is an example of systematic problems being properly solved at the system's level rather than by user hacks.) TeX does have a "non-breaking-space", written "~", but it is used only where TeX cannot do the job by itself --- mainly, after abbreviation periods (where line breaks must be avoided), to distinguish them from sentence periods (where line breaks are preferred).
  While a break between "of" and "$x$" is not nice, it is not that terrible either. A break after "Dr." may be mistaken for end-of-sentence, but a break after "of" will not. Anyway, wikipedia editors and readers have much worse problems to worry about: bad grammar, jumbled order, confusing explanations and even incorrect statements. (One of my "contributions" to this article was a completely wrong value for the roots of $\lambda ^{3}-1$. No one complained about that.)
  Editors should be working on those real problems, not on subtle formatting details; and anything that makes it harder for them to work on those problems, is bad for wikipedia. Won't you agree that the NBSPs make the affected sentences harder to read and edit, especially for novice editors (which are desperately needed to keep the project alive)?
  I did not delete the NBSPs because I did not "like" them, but because they were indeed standing in the way, and I could not see what good they were doing to the article. And I still don't. Since line breaks fall in different places for different readers, most of those NBSPs will have no effect; perhaps only one out of every 20-30 will actually prevent a slightly objectionable line break like
  thus the arrow is parallel to
  $x$, and therefore the vectors
  (but then also cause the previous line to be shorter, which is not nice either). Yet every one of those NBSPs will be visible in the source, standing in the way of editors.
  If there was some way to achieve the same effect without having so much impact on the readability of the source (say, like the "~" of TeX), I would offer to put back your edits, out of respect. But if the only way is to put back the "&nbsp;"s, then no, sorry: I still believe that wikipedia is better without them, and I will not work to make it worse.
  Please, please, consider investing your time into editing articles for contents and clarity, rather than looks. There are literally millions of articles that need such help, and that would really help millions of people out there. All the best, --Jorge Stolfi (talk) 17:31, 5 February 2013 (UTC)

Jorge may wonder, but the revision saved by me 4 days ago does not contain a single cubic equation. Not a single mention of it, so it is unclear how it might suggest a “completely wrong value for the roots of $\lambda ^{3}-1$”. The same for the revision when I edited the article first. Possibly, Jorge fixed such a crap in the past, but why to speak about it now? Does he extort a vaunt? We all fix a crap (at least, me too), and I did not receive a single vaunt for it, but sometime receive insults and hatred. What about &nbsp;s? I just wait a third-party opinion. Arguments of Jorge are not convincing: “TeX already does a very good job at line breaking” and “TeX does have a "non-breaking-space"” are off-topical, and if one feels uneasy with a complicated wiki syntax, then let him/her just avoid making a serious job there. Incnis Mrsi (talk) 19:28, 5 February 2013 (UTC)

• I did not mean to imply that you or NBSPs were in any way to blame for my mistake. I was lamenting to the birds that such a gross factual error went unnoticed until I fixed it myself. It shows how thin is wikipedia's editor base of these days: only five years ago, massive edits in such an important page would have attracted many critical eyes, complaints, and additional edits. And this dispute of ours would have turned into virtual bar-fight. Sigh... --Jorge Stolfi (talk) 23:37, 5 February 2013 (UTC)

I saw the note at the Village pump. My personal preference is to solve both problems at the same time by using the actual character rather than the HTML code. It's option-space on a Mac, or " " if you want to copy and paste. Then you'll have readability benefit of keeping the words connected and not have the intimidating mess in the edit window. WhatamIdoing (talk) 20:17, 6 February 2013 (UTC)

Opening sentence

Is something an eigenvector with respect to a matrix, or a transformation? Currently the article is written primarily with the perspective it is with respect to a matrix. But really it is a property with respect to a transformation (including those defined with matrices). From a pedagogic point of view I understand matrices might be easier than abstract linear transformations, but in this context I think the matrix point of view makes it more difficult to understand what an eigenvector is. So I think the opening sentence should use the term transformation instead of matrix. Mark M (talk) 03:28, 24 February 2013 (UTC)

Well, I do not think that the article is written "primarily with respect to a matrix". It does start that way, but it does give the abstract linear algebra definition too.
I am trying to imagine what sort of reader will look up this topic. As a computer guy, my view may be skewed; but I would guess that the reader will most likely be a technical person (programmer, scientist, engineer) who needs the concept but does not know what it is, or does not quite remember it anymore. I would guess that he understands matrices but not necessarily abstract linear algebra. If his need is related to a practical application, his "linear operator" will be a matrix anyway.
I agree that it is hard to see the importance of eigenvectors in a matrix context; but is the abstract definition really easier to understand? Abstractions are not easy to learn; one must learn at least two concrete examples before seeing the merit of an abstraction. I would think that matrix product is the first concrete example of a non-trival linear operator for most people.
The problem with mentioning lienar operator on the first sentence is that one would have to rewrite it entirely in those terms, and then the matrix view would be lost. Maybe there is a way out but I do not see it... --Jorge Stolfi (talk) 21:49, 24 February 2013 (UTC)

Hm, perhaps we can rearrange things so that the generalizations are mentioned earlier, in the second paragraph. Let me try. --Jorge Stolfi (talk) 21:57, 24 February 2013 (UTC)

OK, I have condensed the first two paragraphs (at the cost of losing the distinction of right/left eigenvector, but that is a nit that few will miss) and swapped another two paragraphs, so that the general definition is now closer to the top of the article. Do you think it is good enough? --Jorge Stolfi (talk) 22:20, 24 February 2013 (UTC)

I just want the first sentence (and paragraph) to be understandable to the widest possible audience. I guess my point is that the word "transformation" has an English meaning that is close enough to the mathematical definition (unlike the word "matrix"), which could be used to define what an eigenvector is. In this sense, a "matrix" is more abstract than a "transformation". Also, it is conceptually more natural to think about eigenvectors with respect to a geometrical transformation (as the pictures in the article suggest), as opposed to thinking about them with respect to a matrix.

That's why I had hoped to first sentence would use the word "transformation"; then later in the first paragraph, to give a better understanding to those who are more comfortable with matrices, say something like "Equivalently, an eigenvector of a square matrix is..". (Of course the unavoidable piece of jargon is the word "vector"..) Mark M (talk) 09:57, 25 February 2013 (UTC)

I see. However, it cannot be any transformation; we would have to say a linear one; and "linear transformation" is not a widely known concept. As for the widest possible audience, I cannot imagine how we could usefully explain eigenvalues and eigenvectors to someone who does not even know matrix multiplication.
Maybe we could write

A square matrix $A$ can be viewed as a linear transformation that maps any column vector $v$ to the vector $Av$. An eigenvector of $A$ is a non-zero vector $v$ that, when transformed by $A$, yields the original vector multiplied by a single number $\lambda$; that is, $Av=\lambda v$. The number $\lambda$ is called the eigenvalue of $A$ corresponding to $v$.<ref name=WolframEigenvector>...</ref>

Would that do? All the best, --Jorge Stolfi (talk) 16:18, 25 February 2013 (UTC)

Why does it have to be linear? Mark M (talk) 17:19, 25 February 2013 (UTC)

And anyway, you are still unnecessarily introducing matrices into the first sentence. Mark M (talk) 17:22, 25 February 2013 (UTC)

• The concept is totally useless if the operator is nonlinear. For example, if F is not linear, and F(v) = λv, it does not follow that F(2v) = λ(2v), or F(-v) = λ(-v). Thus eigenvectors cannot be normalized and do not form eigenspaces. Indeed none of the properties (and applications) of the eigenXX of linear operators that are described in the article would apply to those of a non-linear operator.
  As for matrices: I may have a biased view of the world, but believe that, among all the people who need/use eigenvalues, perhaps 80% or more need/know then in a matrix context, and perhaps 50% would be at least confused by the term "linear operator". Perhaps for mathematicians "linear operator" is more basic/elegant/whatever, but I do not think it is the case for the rest of the world. Linear operator is one level of abstraction above matrices; most people learn things from concrete to abstract, not the other way around. --Jorge Stolfi (talk) 06:09, 26 February 2013 (UTC)

Consider Springer's encyclopedia of mathematics, which does not insist on linearity. It also doesn't mention matrices in the entire entry. "A non-zero vector which is mapped by to a vector proportional to it" is clear, concise, correct, and doesn't require matrices. Also, regarding who is seeing this article, perhaps consider the incoming links. I will also point out that the Spanish featured article doesn't even mention matrices in the lead. Mark M (talk) 09:35, 26 February 2013 (UTC)

Wow, you do hate matrices... 8-)
An entry on eigenvectors that does not mention matrices is not surprising in an encyclopedia of mathematics. (Are you familiar with the Bourbaki books? 8-) That is obviously a book that engineers should not buy...
I fully agree that a definition in terms of operators would be more elegant, and would be the best choice for a mathematics book; but this article does not exist for mathematicians, that is the point.
Now that you menioned it: the so-called "encyclopedias of X" have appropriated the name for something that is not at all like an encyclopedia. They have their role and merits, but Wikipedia defintely must not try to be like them.
Alas, that indeed seems to be happening in many technical areas: a few well-meaning but misguided experts decide to "organize" a subject X by turning all the articles on that subject into an "encyclopedia of X". The results have been disastrous: overly long articles full of formulas and poor on intuition, that use jargon and advanced abstract concepts from line 1, and that only experts can understand -- but which by their size and interconnections are very hard to read and impossible to edit. And, since those editors invariably underestimate the effort needed to write a coherent book, they run out of steam and give up when their "encyclopedia of X" is still is full of holes, inconsistencies and errors.
As for links: first, the vast majority of readers get to articles like this one via Google or the search box. Second, links from advanced math articles are naturally more numerous, but surely the links from elementary math and applied science articles (and there are many of those in the list) are followed much, much more often. (A reader of the "Hamiltonian lattice gauge theory" article is not likely to click on "eigenvalue", is he?) Moreover, this article is included in the navboxes "Areas of mathematics" and "Linear algebra"; so the "What Links Here" tool will stupidly list every article that includes either of those two navboxes, swamping the few articles that actually link to this article in the text.
Finally, another argument for starting with matrices: Eigenvalues are most used with symmetric operators, both because they are important and because they have all-real eigenvalues. Everybody understands what a symmetric matrix is; but in order to define a symmetric linear operator in the abstract one must introduce an inner product or some other non-trivial device.
All the best, --Jorge Stolfi (talk) 16:56, 26 February 2013 (UTC)

PS. The trend I described above is by no means limited to mathematicians: engineers are great offenders too -- and while mathematicians at least value elegance, engineers don't seem to worry much about it either... 8-(

You seem to have strong opinions on this matter that aren't likely to change, so I'm not going to pursue this. But I will correct something you've said: I like matrices quite a lot. :-) Mark M (talk) 17:17, 26 February 2013 (UTC)

Um, er, well, on second thoughts, perhaps I need to take another long vacation from Wikipedia. Sigh. Sorry for all the time made you waste on this nit. All the best, --Jorge Stolfi (talk) 19:08, 26 February 2013 (UTC)

Typo

Surely this:

The matrix A is invertible if and only if all the eigenvalues \lambda_2 are nonzero.

Should read:

The matrix A is invertible if and only if all the eigenvalues are nonzero.

174.124.25.60 (talk) 23:34, 14 April 2013 (UTC)

Math-free introduction?

Could we get one math-free sentence in the introduction? How about this:

If a piece of cloth is stretched, the eigenvector is the direction of stretching, and the eigenvalue is the amount of stretching. 129.219.155.89 (talk) 17:53, 6 March 2014 (UTC)

Yes, and I took liberty to expand it. ᛭ LokiClock (talk) 10:29, 9 March 2014 (UTC)

Illustration in Eigenvalues and eigenvectors#An example

The removal of the illustration and the subsequent revert do highlight a slight problem, at least as I see it. While the illustration does correctly show how individual vectors transform, it introduces the additional and entirely superfluous features of an affine space that only serve to confuse, including a hint that a transformation of the vectors is associated with their position in the affine space and as a set of axes that incorrectly suggests that the affine space has an origin. IMO, it would be distinctly preferable to replace the illustration with vectors in a vector space, by having all the vectors radiating from the origin. —Quondum 06:30, 19 February 2014 (UTC)

The purpose of the illustration is written underneath it. The arrows are not supposed to be eigenvectors and it doesn't say they are. The point is that line segments parallel to eigenvectors don't rotate, but those which are not do rotate. It is an interesting and useful thing for the reader to learn, which would not be illustrated by vectors radiating from the origin. I strongly disagree with "incorrectly suggests that the affine space has an origin", since the image does not show an affine space. It shows the effect of transforming a vector space, whose origin is where the axes intersect as usual. McKay (talk) 07:15, 19 February 2014 (UTC)

Okay, so let's consider the illustration as depicting a vector space. Who said anything about eigenvectors? The caption refers to vectors, and so did I. And since we are dealing with a vector space, what are line segments? You seem to be imbuing a vector space with constructs that have not been defined. And if the vectors are shown radiating from the origin, the eigenvectors would maintain direction, and the vectors that are not eigenvectors would show exactly the rotation that they show now, only without the additional translation. —Quondum 07:42, 19 February 2014 (UTC)

I think the image would be enhanced by showing the eigenvectors as arrows from the origin, and removing the arrowheads from the line segments. Line segments are an elementary concept in vector spaces and will be meaningful even to people who can't write a definition for them, because they correspond to intuition. McKay (talk) 23:54, 19 February 2014 (UTC)

I'd be a lot happier with File:Eigenvectors-extended.gif. That clearly shows the eigen vectors. Also making it so it matched the example in the text ${\bigl [}{\begin{smallmatrix}3&1\\1&3\end{smallmatrix}}{\bigr ]}$ would be nice. Having one of the eigen values 1 also over simplifies things and is a bit of a special case.--Salix alba (talk): 07:18, 20 February 2014 (UTC)

Line segments may be part of Euclidean geometry, and they are an elementary concept, but they are not definable just in terms of vector spaces. They require some form of betweenness geometry. It would be equally elementary to show a rectangle or any image being distorted by the linear map. The orientation of the line segments is unnecessary, and likely to mislead someone who has misunderstood the arrow metaphor to mean a vector is equivalent to an oriented line segment, which is exactly what the author of the image intended to suggest. ᛭ LokiClock (talk) 07:37, 2 March 2014 (UTC)

The elementary premathematical notion of a vector is something with a magnitude and direction. Thus vectors are arrows in Euclidean space, with arrows having the same magnitude and direction but different starting points identified. Of course, with this identification the space of vectors is indeed a vector space. So I don't really think it's a problem that the picture doesn't conform to our usual idea of a vector space as having all vectors emanating from the origin. I think a bigger problem is that which vectors are eigenvectors needs to be made more clear. Currently, the arrows that stand out the most are not the eigenvectors. Sławomir Biały (talk) 13:13, 8 March 2014 (UTC)

I don't think that this article is going to be of most interest to reader with an elementary premathematical level of mathematical sophistication. Nevertheless, let's accept the premise that we are representing displacement vectors from arbitrary points. The diagram then makes the implied association of movement of the starting point with the vector scaling, which is distracting and a burden on the reader to disentangle. This notion would suggest keeping the starting points fixed or have them move randomly rather. Any structure that is suggested but not relevant can lead to confusion. aA case in point: the massive, perennial arguments at Talk:Exponential function that are rooted in an invalid association between two fundamentally different definitions, and this is with people of some sophistication. Even presenting a context in which to interpret the association would help with interpretation, e.g. mentioning stretching of a rubber sheet affecting relative positional displacements. Nevertheless, it is imply an opinion/preference on presentation that I'm expressing; it is not a strong opinion. —Quondum 16:22, 8 March 2014 (UTC)

Quondum, this article definitely has a large premathematical audience - I probably first came to the idea through trying to find out what a quantum state is, and atomic orbital also links here in the elementary discussion, as do many articles on quantum physics topics that are elementary or else prominently referenced in popular science media. Even if it can't be processed well enough to provide new facility to such a person, it makes a lasting impression. The article may only be understood with the implicit understanding that vectors sit in a vector space, for which the meaning of transforming one space into another is to take a linear map. The perception of a valid transformation of a structure is the same as the perception of the structure. Instructing the reader to admit a generalization of one while maintaining a restriction on the other makes this structuralization of the restricted concept unintuitive. It damages the reader's intuition by confusing them as to the rationale of the restriction of fixed points of the map to lie on subspaces through a common origin. What happens when you try to go into the abstract algebra under the impression that vectors start from any location? The unique identity isn't recognizably true, and the concept that the linearity laws should constitute a function being a transformation of spaces is unjustifiable and obviously incomplete. Because the new picture doesn't fit with the old one and obviously match up to it, learning how to work with vector spaces through their axiomatic definition becomes a barrier that can only be resolved through the help of someone who knows the truth, often by accident, because what would the problem be, and how would a premathematical person articulate it? If they are supposed to be able to use Wikipedia math articles at all, and those articles are supposed to be free to depend on references to the treatment by Wikipedia of the concepts foundational to the article, then the concepts as presented in the article must be consistent with those references. Therefore the picture should conform to our usual notion of vector space, because that's the one Wikipedia is representing to the reader. ᛭ LokiClock (talk) 10:04, 9 March 2014 (UTC)

Yes, this echoes what I feel very well. The premathematical audience's "lasting impression" counts when they become no longer quite so premathematical (I guess this is where I meant that it would be "of interest" to them, even if they'd read it earlier), when they start needing to get to actually understand the use of the concept, and will be stuck with this image with its confusing mix. However, most of the editors do not seem to share this concern. —Quondum 16:06, 9 March 2014 (UTC)

To be clear (in case I have misunderstood): are we currently talking about this diff? Cf. a diff from June 2012. N.b. the related discussion. ←←νημινυλι (talk) 09:10, 18 March 2014 (UTC)

The latest one, although the other discussion is relevant for context, since you bring it up. ᛭ LokiClock (talk) 14:40, 19 March 2014 (UTC)

Mangled lede

The second (depending on how you count) paragraph of the lede starts:"If 2D space is visualized as a piece of cloth being stretched by the matrix, the eigenvectors would make up the line along the direction the cloth is stretched in and the line of cloth at the center of the stretching, whose direction isn't changed by the stretching either. The eigenvalues for the first line would give the scale to which the cloth is stretched, and for the second line the scale to which it's tightened. A reflection may be viewed as stretching a line to scale -1 while shrinking the axis of reflection to scale 1. For 3D rotations, the eigenvectors form the axis of rotation, and since the scale of the axis is unchanged by the rotation, their eigenvalues are all 1."

This is virtually useless. 2D space? Is the target audience supposed to know what "2D" means? Are they supposed to know that most 2D spaces (eg. the surface of a sphere) can NOT be "visualized" as a piece of cloth? Are they supposed to know that an eigenvector is not JUST applicable to 2D? Where is it explained how the number of components of a vector relate to the dimensionality of some "space"? Why is a plane, which any kid in high school IS familiar with, called a space? How does a process of "stretching", which is obviously a change with time, have ANY bearing on the linear transformation of a vector? How can "vectors" ← note the plural! make up A LINE ←note the singular ?? I'm not qualified to write on this subject, but its my understanding that there must be as many eigenvectors as the dimensionality of the space. If this is correct, how do you get from a single direction (stretching) to two vectors?? Center of stretching?? What does that mean?? We are supposed to visualize a real object, a piece of cloth, being acted on by a matrix?? Please tell me you know the difference between a real object and a mathematical abstraction! An abstraction isn't going to have ANY effect on a piece of cloth. How do you "tighten" a piece of cloth that is being stretched? What is the meaning of "scale" and how is the cloth stretched to it? I have NO idea how to visualize a reflection being negative stretching, in point of fact, its not, since negative stretching would be compression, wouldn't it? Finally, tossing in a claim about rotations (of what? cloth?) in 3D is less than useful, imho. BTW it is "axes" not "axis" for the plural.

This paragraph is so flawed that either a total rewrite or just complete removal are the only alternatives. It is apparently an attempt to give a concrete example of a linear transformation, but does a really, really bad job of it.173.189.77.242 (talk) 22:23, 14 June 2014 (UTC)

Minor feedback

I would have understood the statement "yields a constant multiple of v" from the first paragraph much sooner if it was clear to me that "a constant multiple of v" implied multiplication by a scalar (or are there more complicated scenarios in which it is not a scalar?) — Preceding unsigned comment added by 71.49.135.68 (talk) 04:15, 10 October 2014 (UTC)

I've reworded it slightly; hopefully it is better now. —Quondum 05:15, 10 October 2014 (UTC)

General definition

Re. zero vectors as eigenvectors: "If we would like the zero vector to be an eigenvector, then we must first define an eigenvalue of T as a scalar \lambda in K such that there is a nonzero vector v in V with T(v) = \lambda v . We then define an eigenvector to be a vector v in V such that there is an eigenvalue \lambda in K with T(v) = \lambda v ."

This makes the zero vector an eigenvector whose eigenvalues are all the eigenvalues of all the other eigenvectors. This seems kluge-ish and poorly motivated. What is the motivation for wanting the zero vector to be an eigenvector? Philgoetz (talk) 16:02, 6 February 2015 (UTC)

Technically, what should be in the article is what is in the sources, and that would be the motivation for including the mention here. I have not read the one reference given, but the natural answer to your question seems to me to be that this allows the set of eigenvectors associated with a given eigenvalue to be an eigenspace, else they would differ by the zero vector. Having the set of eigenvectors associated with an eigenvalue being a vector space can simplify statements about it enough that the quirkiness of the definition may be justified. —Quondum 17:58, 6 February 2015 (UTC)