# Talk:Eigenvalues and eigenvectors

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## 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 eigenvectorsEigensystem – 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.--JohnBlackburnewordsdeeds 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)

## 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 $...$ 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 $tags, but if you do decide to use those, then be absolutely sure the contents of the tag are valid LaTeX code! They should also have consistent typesetting, such as the matrix name being a bold A. For one, in my user preferences, I set it to always render math tags as PNG images. This is why I got to see all the errors and you probably did not. I advise you to temporarily change that setting to always render PNG just to be sure everything is working properly. And of course, double-check everything with the preview function before committing the changes. I haven't had the time to look through all the other edits you've made to the rest of the page yet, but I just wanted to point out that urgent issue. — Kieff | Talk 02:24, 4 February 2013 (UTC) • Indeed there were a couple of things that MathJAX understands but the Wikipedia math-to-PNG renderer does not: "×" instead of \times, \phantom, "…", and some other UNICODE weirdos that I was unable to see but went away once I re-typed the offending formula. I have changed my Wikipedia profile to use the PNG renderer and now the page displays without errors. I agree that the PNG option makes the page rather too heavy; it also renders things in the wrong font size. Is it too unrealistic to expect people to switch to MathJAX yet? It seems to be the way of the future... As for using boldface for matrices and vectors, see the comment above. It not a general convention (mathematicians and physicists do not seem to use it), it is not needed in a text of this size, and it makes editing more painful. I believe the page is now consistent, without bold for matrices and vectors. All the best, --Jorge Stolfi (talk) 03:45, 4 February 2013 (UTC) I do not understand a maniacal addiction of certain users to [itex]. With MathJax, versions of Jorge Stolfi requires about 20 seconds to be rendered in my browser. I think that with PNGs it easily can cause a vomit. Why he uses this resource-consuming [itex] to say “$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 [itex], 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 [itex]. In any case, Wikipedia does not need a human job for conversion from simple {{math}}/{{mvar}} to [itex]. 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) ## 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: 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: 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: 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) ## Destruction of &nbsp; With spaces With &nbsp; 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 [itex]! After edits of Jorge Stolfi all “matrix&nbsp;{{mvar|A}}” (and similar), which I put into the article, became “matrix [itex]A$”. I already said about addiction to $, but what is a substantiation for destruction of &nbsp;? Incnis Mrsi (talk) 07:12, 4 February 2013 (UTC) • Well, sorry for having erased the work you invested in adding the non-breaking spaces. I would gladly put them back if I could be convinced that they are a positive thing. Please read my arguments against their use. As for excessive use of [itex]...$, 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. [itex] .
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 $A v$. 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, $A v = \lambda v$. 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.
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.