Talk:Eigenvalues and eigenvectors

I don't agree with the full set of changes in special:diff/1217830133/1228837393 by @Jorge Stolfi, but I think he has some valuable ideas for changes to the lead which should at least be discussed / taken into consideration instead of summarily rejected. Pinging @XOR'easter, @Tito Omburo, @Chetvorno who chimed in a bit about the lead here previously (I'm probably missing other folks). Here's how Jorge Stolfi left the lead:

In linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector of a linear transformation ${\displaystyle T}$ is a non-zero vector ${\displaystyle v}$ that is only multiplied by some scalar factor ${\displaystyle \lambda }$ when transformed by ${\displaystyle T}$; that is ${\displaystyle T(v)=\lambda v}$. The scalar ${\displaystyle \lambda }$ is the corresponding eigenvalue or characteristic value associated with ${\displaystyle v}$.^[1]

Eigenvalues are often called characteristic roots because they are the roots of a characteristic polynomial associated with the linear transformation. Specifically, if ${\displaystyle M}$ is the square matrix of the linear transformation ${\displaystyle T}$ on some specified basis, the eigenvalues of ${\displaystyle T}$ are the values of ${\displaystyle \lambda }$ that satisfy the equation ${\displaystyle \det(M-\lambda I)=0}$, where ${\displaystyle \det(\cdot )}$ denotes the determinant of a matrix and ${\displaystyle I}$ is the identity matrix with same size as ${\displaystyle M}$.

In two- and three-dimensional geometry, a vector can be depicted as an arrow with a specific length and direction. A linear transformation acts on such vectors by some combination of scaling, rotation, or shearing. Its eigenvectors are those non-zero vectors that are only scaled. The corresponding eigenvalue is the factor by which an eigenvector is scaled. If the eigenvalue is positive, the eigenvector is stretched or shrunk, without changing its direction; if it is negative, the vector's direction is also reversed; and if it is zero, the vector is anihilated.^[2]

Any linear transformation of a space of dimension ${\displaystyle d}$ is completely defined by a set of ${\displaystyle d}$ mutually orthogonal eigenvectors and their associated eigenvalues. For this and other reasons, eigenvalues and eigenvectors are extremely important in all areas of mathematics, science, and technology where linear algebra is used. In mechanics, for example, they arise in the analysis of the rotation of a rigid body, of the behavior of non-isotropic materials under stress, of the vibrations in an elastic structure, and many other phenomena. In statistics, the eigenvectors of the covariance matrix of a set of ${\displaystyle d}$-dimensional data points may reveal the main independent factors affecting those data. In the control of industrial systems, the eigenvalues of the matrix describing the feedback signals can reveal the stability of system under momentary distrurbances.

I think the first paragraph here is unnecessarily less accessible to follow than the previous version, some technical detail can be deferred a bit maybe, and I don't like specifying 2–3 dimensions when vectors can be thought of geometrically in any dimension (albeit mental images get more abstract). But in particular I like the expanded discussion of various applications, which is more concrete and balanced than what was there before. The change from "serve to characterize it" to "completely defined by" is more emphatic though I'm not sure all that different. (Eigenvectors don't have to be mutually orthogonal, but that mistake can be written out.) –jacobolus (t) 01:12, 16 June 2024 (UTC)

It is indeed "less accessible", but necessarily so. The current version is not only incomplete, but wrong.
Take the matrix of rotation by 90 degrees in ${\displaystyle \mathbb {R} ^{2}}$. The matrix has only integer entries (-1, 0,+1) but the eigenvalues are complex, ${\displaystyle \lambda _{1}=+\mathbf {i} }$ and ${\displaystyle \lambda _{2}=-\mathbf {i} }$. The eigenvector of the first one is ${\displaystyle (1+\mathbf {i} ,1-\mathbf {i} )}$, which is changed to ${\displaystyle (-1+\mathbf {i} ,1+\mathbf {i} )}$. So, what is the "direction" of this eigenvector, an in what sense is it preserved?
Again, the first paragraph of an article on "Foo" must define the concept "Foo" precisely and completely. It must be such that everything that is "Foo" fits the definition, and anything that is not "Foo" does not fit it. One must avoid unnecessary jargon, yes -- but on advanced topics, like this one, the definition must assume knowledge of whatever basic stuff that is needed to achieve that goal.
As for eigenvectors not being always orthogonal, that is true, but note that I wrote "by a set of ${\displaystyle d}$ mutually orthogonal eigenvectors", not "by the ${\displaystyle d}$ eigenvectors.
All the best, --Jorge Stolfi (talk) 22:07, 16 June 2024 (UTC)

In this case there are no real-valued eigenvalues or eigenvectors, and no preserved (real) vector directions. If you expand your domain to complex numbers there are preserved complex directions (using a suitably generalized concept of "direction"). There's some discussion about this general topic in the article, but it's not particularly clearly organized, complete, or accessible; ideally we would do better, maybe by more completely discussing all of the possibilities for 2-dimensional real-valued matrices with more pictures. –jacobolus (t) 22:50, 16 June 2024 (UTC)

But those are the eigenvalues and eigenvectors of that matrix. A lede that implies that eigenvalues are real numbers is just wrong. Jorge Stolfi (talk) 14:03, 30 June 2024 (UTC)

This is really down to definitions and point of view. If you consider your real-valued matrix to represent a linear transformation of a vector space over the scalar field R, then there are arguably no eigenvalues or eigenvectors in this circumstance. If you consider your real-valued matrix to represent a linear transformation of a vector space over the "scalar" field C, then there are eigenvalues and eigenvectors, and they represent a generalized notion of scaling and preservation of direction. –jacobolus (t) 21:22, 1 July 2024 (UTC)

It is not the case that there is always a set of d mutually orthogonal eigenvectors for every matrix of dimension d. (Eigenvalues and eigenvectors are a non-metrical vector-space concept, while orthogonality is a metrical concept. You might be thinking of the SVD or the like.) –jacobolus (t) 23:31, 16 June 2024 (UTC)

Either lede seems fine to me, modulo the detail about orthogonality. Stofli's has the advantage of being a more thorough summary, but the disadvantage of being less elementary. Tito Omburo (talk) 10:58, 16 June 2024 (UTC)

Please see my reply above... --Jorge Stolfi (talk) 22:07, 16 June 2024 (UTC)

The words in in the style guide are clear. For the Lead section:

• we should favour understandability over completeness
• we should make the lede accessible to as broad an audience as possible
• we should cultivate interest in the rest of the article
• We should show how the topic is useful and interesting
• We should try to avoid difficult-to-understand terminology, symbols, mathematical equations and formulas, and where that's not possible, we should provide context and simple definition

Chumpih ^t 13:52, 1 July 2024 (UTC)

Here, here! I completely agree. Precision and nit-picking belong in an appropriate section of the article, if anywhere, but not in the lead. Zaslav (talk) 23:26, 7 July 2024 (UTC)

First, the style guide is only the opinion of those editors who wrote it.
Second, it does not say that. It says "The lead section should briefly summarize the most important points covered in an article in such a way that it can stand on its own as a concise version of the article." And also "Where uncommon terms are essential [which is the case here!], they should be placed in context, linked, and briefly defined."
It is unfortunate that in some countries the school system only manages to instill students with hate for mathematics. Or for science in general. The attempt to make articles accessible to those readers, by replacing mathematical definitions with "simplified" definitions that will not "scare" them, it thoroughly misguided. It will not tell them what the subject is, but give them a false idea of it while letting them think that they understand. It will not reduce ignorance, but only deliver ignorance masquerading as knowledge. Jorge Stolfi (talk) 15:06, 9 July 2024 (UTC)

When you say 'it does not say that', could you define 'that'? is there a particular one of the bullet points that is not supported by the text at WP:LEDE? I see you quote some of the sentences from LEDE - there are other sentences there, too. Re. WP:MOS, it represents the rules by which we should write articles; it's more than just 'the opinion of those editors who wrote it' - there's hard-won consensus here, as may be seen from the thousands of archived debates at WT:MOS. Partial knowledge isn't necessarily fallacious. Much of the skill here is to make new topics accessible, perhaps by example, perhaps by gentle introduction. Since we're quoting now: Make the lead section accessible to as broad an audience as possible. Chumpih ^t 16:04, 9 July 2024 (UTC)

Some important, frequently used terms are missing from the matrix section; two I can think of offhand are: index of a matrix (not the indices (i,j) of an element), spectral radius, others I'm forgetting. These terms are also missing from Matrix (mathematics) and Square matrix. This is serious. Zaslav (talk) 23:25, 7 July 2024 (UTC)

Wondering if it's still necessary to have the too technical banner for the geology and glaciology section. I tried my best to make it easily digestible but I didn't want to remove the banner myself. Curious to hear another opinion. Mgcaptainzanko (talk) 06:42, 14 December 2024 (UTC)

Thanks for your edits and for asking for feedback!

I think the keys for these kinds of sections are 1) clearly connect to the article topic, 2) introduce critical aspects of related field. I think the second paragraph addresses the connection reasonably well.

A "clast's fabric" is not something familiar (I guess) to most readers outside of geology experts, but it is the main subject in the second paragraph. So a couple of sentences describing "clast" and "fabric" would be a huge improvement. Is "clast" essential? Fabric is easy to understand but "clast" is a completely new word for me.

The first paragraph has what I consider superfluous or at least puzzling content. "In the field, ..." to the end of the paragraph seems to be some kind of details not directly related to eigenvalues. I would replace that with more on "clast fabric". The second paragraph starts abruptly "The output for the orientation tensor...": what tensor?

HTH Johnjbarton (talk) 16:24, 14 December 2024 (UTC)

I've been using Google's Gemini 2.5 Pro Experimental large language model to create summaries for the most popular articles with {{Technical}} templates. This article, Eigenvalues and eigenvectors, has such a template in the "Geology and glaciology" section. Here is the paragraph summary at grade 5 reading level which Gemini 2.5 Pro suggested for that section:

Scientists who study rocks and glaciers look at how small pieces of rock are arranged in the ground, especially in soil left by glaciers. To understand this better, they use special math tools called eigenvectors and eigenvalues. These tools help take information from hundreds or thousands of rock pieces and turn it into just a few numbers. These numbers tell the scientists the main directions the pieces are pointing and how tilted they are. This helps them figure out if the pieces are all mixed up, lying mostly flat, or lined up in one main direction, and they can draw special pictures to show this.

While I have read and may have made some modifications to that summary, I am not going to add it to the section because I want other editors to review, revise if appropriate, and add it instead. This is an experiment with a few dozen articles initially to see how these suggestions are received, and after a week or two, I will decide how to proceed. Thank you for your consideration. Cramulator (talk) 12:20, 2 April 2025 (UTC)

This proposed text is, frankly, garbage. It is both incredibly vague and also seems incorrect or at least quite misleading. Please don't use AI tools to write any part of Wikipedia. –jacobolus (t) 12:44, 2 April 2025 (UTC)

Please read WP:Verify. These content snippets are useless without sources. Johnjbarton (talk) 16:10, 2 April 2025 (UTC)

This is the dumbest bullshit I have ever read. Tito Omburo (talk) 22:14, 2 April 2025 (UTC)

I am retracting this and the other LLM-generated suggestions due to clear negative consensus at the Village Pump. I will be posting a thorough postmortem report in mid-April to the source code release page. Thanks to all who commented on the suggestions both negatively and positively, and especially to those editors who have manually addressed the overly technical cleanup issue on six, so far, of the 68 articles where suggestions were posted. Cramulator (talk) 22:12, 4 April 2025 (UTC)

I removed a subsection which seems wrong. It does not agree with the cited source in a clear way. If it can be restated in a more normal way using ordinary linear algebra notation, with a source, it can be restored. But the use of bra-ket notation appears to obfuscate any mathematical content therein. Tito Omburo (talk) 22:04, 27 July 2025 (UTC)

The article Frobenius covariant gives the relationship ${\textstyle F_{i}(A)=v_{i}w_{i}}$, where ${\textstyle v_{i}}$ and ${\textstyle w_{i}}$ are the right and left eigenvectors of ${\textstyle A}$: ${\textstyle Av_{i}=\lambda _{i}v_{i}}$ and ${\textstyle w_{i}A=\lambda _{i}w_{i}}$. For a Hermitian matrix ${\textstyle A}$, ${\textstyle w_{i}=v_{i}^{\dagger }}$, in which case ${\textstyle F_{i}(A)=v_{i}v_{i}^{\dagger }}$. By taking the α-th diagonal element of the Frobenius covariant, we arrive at $${\displaystyle |v_{i\alpha }|^{2}=(F_{i}(A))_{\alpha \alpha }.}$$ This is equation (3.3) of the paper https://arxiv.org/abs/2306.10638. This equation is an alternative to the equation of subsection Eigenvector-eigenvalue identity of the main article.

The next step: $${\displaystyle v_{i\alpha }={\frac {v_{i\alpha }v_{i\gamma }^{*}}{v_{i\gamma }^{*}}}=e^{i\phi _{i\gamma }}{\frac {v_{i\alpha }v_{i\gamma }^{*}}{|v_{i\gamma }|}}=e^{i\phi _{i\gamma }}{\frac {v_{i\alpha }v_{i\gamma }^{*}}{\sqrt {v_{i\gamma }v_{i\gamma }^{*}}}}=e^{i\phi _{i\gamma }}{\frac {(F_{i}(A))_{\alpha \gamma }}{\sqrt {(F_{i}(A))_{\gamma \gamma }}}}.}$$ This is equation (3.4) of the above paper. It gives eigenvectors up to the phase factor and is more general than the first equation and the equation given in the subsection Eigenvector-eigenvalue identity. LevTcdd.jl (talk) 18:34, 28 July 2025 (UTC)

What is your point?—Anita5192 (talk) 18:51, 28 July 2025 (UTC)

This is related to the removal noted in the previous thread. Although, having seen it now in a clearer form, am pretty sure it doesn't belong here, among other things being WP:UNDUE weight, based on a single paper. Also, apparent lack of content relevant to the article. Tito Omburo (talk) 19:56, 28 July 2025 (UTC)

It is accepted to have uniform notations in single article, so Dirac's bra-ket excellent notations are inappropriate when another notations are already used.

The Frobenius covariant is defined not to be overlooked through the eigenvalues of the matrix. The above equations determine the eigenvectors of the matrix. So, these equations match accurately the content of the article. LevTcdd.jl (talk) 20:26, 4 August 2025 (UTC)

I don't object to the inclusion of a discussion of the projections onto eigenspaces. But this content ain't that. Tito Omburo (talk) 21:01, 4 August 2025 (UTC)

In the Section of "Eigenspaces, geometric multiplicity, and the eigenbasis for matrices" of the Wikipedia page "Eigenvalues_and_eigenvectors", there is a sentence "Because a nullspace is a linear subspace of the domain". To understand what the domain here is, I think that an above sentence "On one hand, this set is precisely the kernel or nullspace of the matrix (A − λI)." in the same section need to be modified such that the matrix (A − λI) is mentioned as a linear map. With this, which domain in the issued sentence is clarified (the domain of the linear map represented by this matrix).

I made this modification on 2025-09-12 (2025-September-12), but it was reverted by an unexplained reason, so I would like to discussion this topic here. Was my modification valid? Goodphy (talk) 20:21, 13 September 2025 (UTC)

Not sure what the issue is with what was there before. I reverted to the original version, prior to your edit, with an explanation, but I am happy to expand. "Nullspace of a matrix" is almost always introduced before linear transformations, and eigenvalues often are as well (e.g., via Markov chains). The concept of a "linear transformation" is somewhat extraneous here, no? (The domain is just all column vectors, maybe in C^n? Which is clearly asserted.)

The appropriate place for this content would have been in the section on linear transformations. E.g., . Tito Omburo (talk) 20:36, 13 September 2025 (UTC)

It seems that there were places, above the issued section, where linear transformations (linear maps) are mentioned, so you think that the "domain" in this section can be naturally interpreted if people read from the above.

I prefer to have sections that are less dependencies on the above sections for easy readability and human winning against AI summary, but now I respect your point of view. Goodphy (talk) 06:09, 14 September 2025 (UTC)

As far as I remember, I have not encountered any definition of eigenvectors of n-by-n matrices which assumes that their length is n. Further, with this condition, the eigenvectors corresponding to a given eigenvalue of an n-by-n matrix together with 0 would not form a vector space. Therefore, I suggest the omission of the assumption on the length from the definition of eigenvectors of matrices. ~2025-32772-70 (talk) 13:17, 11 November 2025 (UTC)

Done.—Anita5192 (talk) 16:48, 11 November 2025 (UTC)

This is the perpetual awkward problem of what name to use for the number of entries of a column vector / the number of rows and columns of a matrix. Here "length" was supposed to be clarifying to the reader that v was specifically an n-by-1 column vector (as opposed to some other kind of vector, or a column vector with some other number of entries), not to any geometric notion of length. I do not have a strong opinion about whether such a reminder is necessary in this location. --JBL (talk) 18:17, 11 November 2025 (UTC)

I was thinking the same thing. The editor may have meant the number of components, but I didn't think it was necessary to specify that. I think most readers would understand it.—Anita5192 (talk) 18:45, 11 November 2025 (UTC)