Talk:Eigenvalues and eigenvectors

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 [1]. 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)

"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)