Talk:Eigenvalues and eigenvectors

Too technical?
I removed the Technical template. Yes, the subject matter is technical mathematics, but I do not see how anyone can understand the subject matter without first understanding concepts like field, scalar, vector space, vector, linear algebra, and linear transformation, all of which are linked in the lead. Explaining them in depth here would be tedious and redundant. Several knowledgeable editors have been painstakingly refining this article for months to make it more readable, and I believe its present state is very readable for readers who already understand the aforementioned concepts.—Anita5192 (talk) 21:48, 8 February 2019 (UTC)
 * Maybe, the less technically prepared get a less steep introduction by replacing the current first sentence with something like
 * Just a suggestion. Purgy (talk) 08:58, 9 February 2019 (UTC)
 * Just a suggestion. Purgy (talk) 08:58, 9 February 2019 (UTC)


 * i could not agree more! after reading the lede i came to the talkpage to say that this is an utterly confusing way to explain for the reader (only) knowing what 'vector' means that the addititon of 'eigen' to the expression is simply meaning that the vector changes only in its length but not in its direction. so yes, the lede is way too technical. it should explicitly say in the very beginning that the eigenvector of v is any v' that only diifers in length from v but is not rotated to point to another direction. (okay, add, that flipping direction 180 degress by multiplying with a negative value does not count as rotation.)
 * the introduction of all other technical terms BEFORE getting to this simple point is making it too technical. 89.134.199.32 (talk) 20:53, 3 September 2019 (UTC).


 * I have moved the formal definition from the lead into its own section in the body of the article. Hopefully this will resolve the aforementioned issues.—Anita5192 (talk) 23:35, 3 September 2019 (UTC)
 * Totally respect that the formal understanding will require some related concepts. However if you were trying to explain this to a random person you might say eigenvectors and eigenvalues are like ways to refer to the direction and amount something is stretched, like an image, or more properly, a set of data points all undergoing some uniform transformation, then quickly caveat that the formal definition involves some technical restrictions where a grounding in linear algebra would be helpful. Talking about transformations people already encounter, like stretching something, could provide a quick foothold for nonexperts. Just an idea. --173.197.42.83 (talk) 02:32, 20 December 2022 (UTC)


 * As a non-expert reading through a variety of mathematics articles on Wikipedia, this was the only article I encountered where I actually understood what the subject was on the first read of the introduction. I would say it's pretty good, most of the intros are full of jargon and overly precise technical definitions. 74.105.139.28 (talk) 23:14, 7 April 2024 (UTC)
 * Thanks! There was a group effort to improve this in November. See . –jacobolus (t) 02:03, 8 April 2024 (UTC)

Please folks, Wikipedia is not a textbook. It is a reference resource, like an encyclopedia, dictionary, or technical manual: a place where people come to know the definitions and basic facts about some topic. It is fine to make the articles more accessible and understandable, as long as one keeps those readers in mind. So, for example, the lead of an article about "polynomial" should assume that the reader only knows the concept of "function" and the very basic of algebra; but the lead of an article about "eigenvectors and eigenvalues" should assume that the reader at least knows what are vectors and linear transformations, or matrices and matrix products. Trying to make the lead (or the article) understandable to people without that basic knowledge is futile and a disservice to those readers to wohm the article should be directed. --Jorge Stolfi (talk) 13:50, 13 June 2024 (UTC)


 * IMHO the changes made on 2024-06-13 are not beneficial. Per WP:NOTTEXTBOOK: the purpose of Wikipedia is to summarize accepted knowledge, not to teach subject matter.  The previous start was far more helpful in this regard.  Per WP:BRD, I'll revert these changes, and the discussion can continue here.  Chumpih  t 20:19, 13 June 2024 (UTC)
 * The lead of an article must precisely and correctly define the subject. Accessibility does not justify giving an incorrect or very incomplete definition. In particular, the concept of "direction" does not make sense if the eigenvalue is complex (which is often the case, even for real 2x2 matrices). And that quote from WP: is saying that articles should not be written as tutorials for beginners. Jorge Stolfi (talk) 20:42, 13 June 2024 (UTC)
 * That's not so. Per WP:LEDE:  It gives the basics in a nutshell and cultivates interest in reading on ... It should be written in a clear, accessible style with a neutral point of view.  Chumpih  t 20:54, 13 June 2024 (UTC)

Suggested re-ordering
Wikipedia is intended to be a general-purpose encyclopaedia. This marks a contrast with a technical manual for the already expert practitioner.

But in this article, the lead is deeply technical, followed by a section "formal definition" which likewise is deeply technical. The intended readership is left floundering in meaninglessness.

We need the article to arrive very quickly at a general overview. The section "overview", especially with its pictorial illustration, would be much better placed earlier. So I propose reversing the order of "formal definition" and "overview". (Perhaps other minor adjustments might become necessary, but they are second-order effects.) Unless there is serious objection, I propose doing this in about a week (21 June 2021). Feline Hymnic (talk) 16:12, 14 June 2021 (UTC)


 * I disagree. The definition section says little more than what is in the lead, but in more precise mathematical terms. The overview section describes applications which should be preceded by a formal definition.—Anita5192 (talk) 16:33, 14 June 2021 (UTC)


 * I disagree too. Wikipedia is not exactly a manual, but, as an encyclopedia, it is much closer to a manual than to a textbook: a reference resource, where people come to seek answers to specific questions -- not to learn a particular broad topic like "linear algebra", or for mere entertainment. The stucture of the article is indeed awful, but because it tries to be like a textbook. Or many textbooks, each editor adding another pedagogical explanation with a different approach... --Jorge Stolfi (talk) 13:50, 13 June 2024 (UTC)

Zero vector as an eigenvector removed.
The discussion about zero vector being an eigenvector was confusing. The source [1] cited said nothing of the kind, and there is a general consensus among mathematicians consistent with the rest of the article.

I've actually checked the reference provided and there is nothing about eigenvectors at the referenced page (p. 77). The chapter about eigenvectors from that book is actually freely available [link ] and it nicely explains the philosophy of eigenvectors as invariant sub-spaces. All the definitions in the book are consistent with excluding the zero vector as an eigenvector.

[1] Axler, Sheldon (18 July 2017), Linear Algebra Done Right (3rd ed.), Springer, p. 77, ISBN (Links to an external site.) 978-3-319-30765-7 — Preceding unsigned comment added by Ormulogun (talk • contribs) 15:56, 10 February 2022 (UTC)
 * Indeed the zero vector is never an eigenvector. Not even when some eigenvalues are zero. --Jorge Stolfi (talk) 11:54, 13 June 2024 (UTC)

Equation numbers are difficult to find.
Was reading the article, and the line starting "Equation (2) has a nonzero solution..." and couldn't find the referenced equation.

The equation numbers are on the extreme far right side of the page, and as such are a) difficult to find, and b) difficult to associate with their equation when several equations are listed vertically. It's especially a problem with this article because all equations are fairly short.

Is there some way of moving the equation reference numbers closer to the actual equations, on pages where the equations are short?

(I'd venture to guess that nearly all equations in all math pages are short, relative to the width of modern monitors.) — Preceding unsigned comment added by 64.223.87.47 (talk) 20:52, 22 August 2022 (UTC)


 * As far as I know, there is no way to adjust the way this feature positions the equation numbers. If you are viewing on a large screen, you could make the window narrower.—Anita5192 (talk) 23:09, 22 August 2022 (UTC)


 * There should be no equation numbers! A wikipedia article is not a technical article or textbook. Each section should stand on its own, because the target Wikipedia readers come to an article to get the answer to a specific question, and thus are likely to go to a specific section without reading the rest of the article. --Jorge Stolfi (talk) 13:50, 13 June 2024 (UTC)
 * It doesn't seem to me like the equation numbers are too important here. Taking them out would take a couple of changes to the wording of particular sentences, but it would be pretty straightforward if anyone wants to try. –jacobolus (t) 23:56, 15 June 2024 (UTC)

Notation for determinant
Some sections in the article use bars for the determinant, even when applied to a symbolic expression, i.e.
 * $$ | A - \lambda I | $$ rather than $$ \det(A-\lambda I) $$.

This is IMO wrong; vertical bars for determinant is a shorthand in which the bars replace the parenthesis delimiting an matrix given in terms of its elements:
 * $$ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$.

130.243.94.123 (talk) 12:39, 21 August 2023 (UTC)


 * — Anita5192 (talk) 15:11, 21 August 2023 (UTC)

Short description
The current Short description is excessive and my attempts to find a suitable replacement were rejected. Please see WP:SDSHORT and WP:SDNOTDEF and suggest possible alternative Short descriptions. I have removed the current SD to suppress the errors pending a viable alternative — GhostInTheMachine talk to me 19:10, 16 November 2023 (UTC)


 * I don't think you need to get too hung up on 'and', just focus on vectors. Jargon is also not too big of an issue since every reader will know that "eigen" is a jargon-y thing. Suggestions:
 * Vectors only scaled by linear transformations
 * Characteristic vectors of linear transformations
 * Johnjbarton (talk) 19:24, 16 November 2023 (UTC)
 * If the vectors are the "main feature", then perhaps the article should also be renamed to Eigenvectors? — GhostInTheMachine talk to me 19:36, 16 November 2023 (UTC)
 * seems to be OK. It seems to make more sense than references to Characteristic vectors. — GhostInTheMachine talk to me 20:06, 16 November 2023 (UTC)
 * This version is confusing to the point it's unhelpful, IMO. –jacobolus (t) 20:12, 16 November 2023 (UTC)
 * The current short description is fine. "snotdef" doesn't mean the short description can't define the subject, only that it need not do so. Consider the places where short descriptions appear and what would be most useful for readers to read there. For example, they get used in search results pages, related article lists, see also sections, and so on. –jacobolus (t) 19:42, 16 November 2023 (UTC)
 * As an aside, could you please skip past the revert war stage in similar future cases? –jacobolus (t) 19:44, 16 November 2023 (UTC)
 * The current Short description is way too long — — about twice as long as it should be. Please suggest sensible alternatives — GhostInTheMachine talk to me 19:50, 16 November 2023 (UTC)
 * The current one is entirely fine (only 79 characters). If you don't have a better proposal, please go find someone who both cares about short descriptions and also understands enough about the topic to contribute here. –jacobolus (t) 20:00, 16 November 2023 (UTC)
 * Exactly! Understand the subject matter before editing. This is a mathematics article, and as such, the short description must first and foremost be accurate. The subject of eigenvectors and eigenvalues is one of those mathematics concepts that cannot be defined accurately in six words.—Anita5192 (talk) 20:09, 16 November 2023 (UTC)
 * Short descriptions are not definitions. Johnjbarton (talk) 21:19, 16 November 2023 (UTC)
 * I've changed it to Quantities that together describe a linear map. To me, this seems like a reasonable thing to have appear in the places mentioned above, and it doesn't slight one half in favor of the other. XOR&#39;easter (talk) 21:14, 16 November 2023 (UTC)
 * Too general. There are many quantities that describe a linear map. For example, the elements of the matrix corresponding to a linear map of a finite-dimensional vector space.—Anita5192 (talk) 21:18, 16 November 2023 (UTC)
 * Short descriptions do not need to be unique. They need to be short! Johnjbarton (talk) 21:19, 16 November 2023 (UTC)
 * So? We're not trying to fit a whole definition into it. XOR&#39;easter (talk) 21:20, 16 November 2023 (UTC)
 * Something like Mathematical concept linking vectors and matrices is more in the spirit, IMHO. Chumpih  t 22:41, 16 November 2023 (UTC)
 * This one seems nonsensical to me. Eigenvectors don't "link" anything. –jacobolus (t) 23:15, 16 November 2023 (UTC)
 * Yes, good choice, because the spirit of the short description is the help readers place the title in the entire universe of all titles. A reader unaware of "eigen" or of "linear transformation" should learn from the short description that it is first and foremost mathematical, that it is a concept; readers with a bit more math will learn that the arena of math involves vectors and matrices. That is enough.
 * It's pointless to say "linear map", these are not words that mean anything to the intended audience for short descriptions. Johnjbarton (talk) 23:41, 16 November 2023 (UTC)
 * Why not just say Mathematical concept, then? The title of the article itself already has vector in it, after all. XOR&#39;easter (talk) 02:26, 17 November 2023 (UTC)
 * Not unreasonable, but perhaps a little more required... Chumpih  t 09:56, 17 November 2023 (UTC)
 * I don't get it. What does linking vectors and matrices mean? XOR&#39;easter (talk) 02:24, 17 November 2023 (UTC)
 * perhaps Mathematical concepts involving vectors and matrices to then? Chumpih  t 10:11, 17 November 2023 (UTC)
 * Since vectors are in the title, maybe "Mathematical concepts involving matrices" or "Mathematical concepts involving transformation matrices" or "Mathematical matrix concepts" Johnjbarton (talk) 16:48, 17 November 2023 (UTC)
 * Could do. Was sort of thinking that the word 'vector' doesn't appear un-prefixed, so spelling it out isn't overly tautological.  But any of these suggestions seem along the right tracks.  Chumpih  t 17:23, 17 November 2023 (UTC)

I don't believe there is any way to describe eigenvectors and eigenvalues accurately in the small space of a short description. Per WP:SDCONTENT, "short descriptions are meant to distinguish an article from similarly-named articles in search results, and not to define the subject." Hence, I think we should be content with something short and general, instead of attempting to fit completeness into a small space. I propose "Concepts from linear algebra."—Anita5192 (talk) 14:25, 18 November 2023 (UTC)


 * Also a fine choice. Johnjbarton (talk) 15:37, 18 November 2023 (UTC)
 * OK – No objections or further suggestions — GhostInTheMachine talk to me 20:28, 23 November 2023 (UTC)
 * I think it would be more productive to have e.g. "In linear algebra, quantities characterizing a linear map." –jacobolus (t) 20:59, 23 November 2023 (UTC)
 * per 's suggestion, Concepts from linear algebra sounds OK to me. Chumpih  t 22:47, 23 November 2023 (UTC)

Not all vectors have direction
I see a lot of people adding the notion of "unchanged direction". Not all (general vector space) vectors have direction. Think functions & linear differential operators, etc. Ponor (talk) 13:46, 17 November 2023 (UTC)


 * In these cases, the concept of "direction" is more abstract, just like the vector spaces themselves, but I would argue it is still meaningful as a concept. Vectors which are scalar multiples of each-other have the same "direction", and in e.g. the example of Hilbert spaces we can quantify how far apart these directions are using some abstracted version of the concept of angle measure, even when the vectors involved happen to be infinite dimensional. [A quick search turns up e.g. Deutsch (1995) "The Angle Between Subspaces of a Hilbert Space": "The notion of the 'angle' between a pair of subspaces in a Hilbert space is a fruitful one. It often allows one to give a geometric interpretation to what appears to be a purely analytical or topological result."] I think it's important to mention a notion of "unchanged direction" at the top for accessibility to a broad audience. Not everyone knows what scalar multiplication means, and trying to unpack several semesters of undergraduate math courses into the lead paragraph of this article is untenable, so we need to quickly give readers some images to hold onto instead of only providing a fully general version in abstract mathematical notation. If you like I can try to add some sources for this definition. Searching in Google scholar search turns up literally thousands of papers and book chapters (from a wide variety of fields, including some talking about more abstract vector spaces) where an eigenvector is described as a vector whose "direction is unchanged", "direction is fixed", "direction is preserved", "direction is not changed", or similar. I'm sure with some effort screening them we can find a few which would be generally useful surveys for a newcomer reader to take a look at. I want to expand and slightly reorganize the first few sections after the lead to first include a geometric/visual overview and more clearly relate it to finite-dimensional matrix arithmetic, but it would also be good to have some introductory description somewhere near the top about more general kinds of vector spaces, what their linear transformations look like, why we want to know their eigenvectors, etc. –jacobolus (t) 19:31, 17 November 2023 (UTC)
 * "Scaled version of itself" is still better than that "unchanged (very abstract) direction". In some languages people like to say that a vector (in geometry) has a magnitude, direction and orientation, but I don't think that's the case in English. So does a negative eigenvalue mean a change/flip of direction or not? There's an example with complex eigenvalues, what do they do to a 'direction'? Ponor (talk) 07:24, 18 November 2023 (UTC)
 * "Scaled version of itself" is not accessible to as wide an audience as "unchanged direction", because not everyone knows what it means to "scale" a vector. But you'll notice that the current third sentence explicitly says "scaled by a constant factor" for anyone curious about what "unchanged direction" means.
 * Complex eigenvalues of a real-valued matrix imply that there are no (real) eigenvectors. If you want to get to complex-valued vectors, then you need to again abstract your idea of what you consider "direction" or "scaling" to mean. In my opinion making sense of these concepts in a more abstracted way is not a significant obstacle for people who already understand linear transformations over complex vector spaces.
 * In my opinion the upshot of your suggestion/critique is that in order to be extraordinarily pedantic, we should make the article gratuitously exclusive of anyone who doesn't already know the subject at a high level, including e.g. undergraduate students encountering eigenvectors and eigenvalues in class for the first time.
 * magnitude, direction and orientation – this would depend on whether you consider a "direction" to refer to lines or "oriented lines". The words "attitude", "direction", "orientation", "sense", "bearing", etc. are used imprecisely and often interchangeably in English. Making these concepts precise requires formally defining them, but that's not really the point in the context of a few informal sentences here intended to give readers the right basic idea without too much technical overhead. –jacobolus (t) 07:32, 18 November 2023 (UTC)
 * Reading this again, my wording here is harsher than intended. To be clear, I don't think anyone's trying to make articles harder than necessary for less-technical readers. I just think we should be careful about how we trade off between fully formal specifications and informal plain-language descriptions. Precision has been fruitful for mathematics, but it also makes the subject difficult and intimidating. In my opinion one of the most important things we can do at Wikipedia, as the canonical source returned by search engines etc., is to help draw in a wide audience and help people make sense of the purpose, high-level context, and technical people's internal metaphors/concepts for various subjects. That can be in addition to, rather than at the expense of, precise formal/symbolic statements. –jacobolus (t) 17:02, 18 November 2023 (UTC)
 * Oh, please! Precision has not "been fruitful in mathematics": mathematics IS precision! It is the whole point of it. There is no such thing as "simplified" or "intuitive" or "plain language" version of mathematics. It is sad that many students in some countries come out of high school not only hating math but also convinced that they don't need math and have the "right" to not know it.  However, faking the definition of a math concept until it does not scare those readers renders them a big double disservice.  First, it leaves them with the wrong notion of what the concept is, but with the illusion that they understood it. Second, it leaves them with the idea that one can understand and use mathematics with "intuitive" and imprecise definitions and reasonling. That is, it only reinforces that sad state of math education... Jorge Stolfi (talk) 15:21, 9 July 2024 (UTC)
 * If, across the technical parts of the encyclopedia, we only ever write descriptions that are completely and utterly nonsensical to the vast majority of the people looking things up, that is contemptuous of readers and an abject failure at Wikipedia's mission. Endeavoring to always write a description in the plainest language we can makes articles useful to at least 10x as many readers, often 100x as many. Starting with a conceptual plain language description is not "faking" anything, and such descriptions are incredibly helpful even to technical readers. Authors of mathematical and other technical works are in other contexts writing for a very niche specialist audience, and even when talking to themselves often forget to mention (or sometimes never figure out!) what their tools and structures are really about, downplaying or leaving out the context, purpose, and meaning in favor of incidental details of technical formalisms. –jacobolus (t) 16:13, 9 July 2024 (UTC)

Potentially ambiguous pronoun in lede
The clause in question is "The eigenvectors and eigenvalues of a linear transformation serve to characterize it" within the lead section. The problem is that a pronoun typically refers to the subject of the sentence, which in this case is 'the eigenvectors and eigenvalues'. But here the pronoun refers to the object (the linear transformation), differentiated solely by a match in plurality between the pronoun and the subject (the pronoun being singular, matching the object, and the subject being plural). This is jarring. A better form could be The eigenvectors and eigenvalues of a linear transformation serve to characterize that transformation. Is there any advantage to keeping the troubling pronoun? Chumpih t 23:18, :30, 15 June 2024 (UTC)


 * A pronoun can refer to any obvious antecedent, in this case "transformation"; it would be entirely nonsensical for "it" to refer to "eigenvectors and eigenvectors", and nobody is ever going to be confused by this. Making sentences more awkward for the sake of misplaced pedantry just makes writing clumsy and harder to read. –jacobolus (t) 23:38, 15 June 2024 (UTC)
 * Indeed, and as you state, a pronoun can refer to any obvious antecedent ... and therein lies the ambiguity.  Removing this ambiguity is likely to help people trying to parse the sentence.  For sure, the edit to remove the 'it' was made because the 'it' was troubling.  Stating "nobody is ever going to be confused by this" is assumptive.  Chumpih  t 23:57, 15 June 2024 (UTC)
 * In general, Chumpih, your supposed copyedits here are significant regressions, a mix of making things less idiomatic/grammatical and gratuitous (incorrect) meaning changes. Please don't do more copyediting, it's not helpful. If you ever do decide to do more such edits, here or elsewhere, please carefully think about what the version says before vs. after, try to write edit summaries which accurately characterize the changes you are making, and don't tick the "minor edit" box. These changes were not appropriate to list as minor edits. –jacobolus (t) 23:41, 15 June 2024 (UTC)
 * Can you help with a definition of 'significant'? Indeed, we should all think carefully about our edits - I wholeheartedly agree with that.  'Minor edit' is usually indicated for a change which doesn't change the semantics of the article, at least that was my understanding.  Is there a different meaning that should be applied?  Chumpih  t 00:02, 16 June 2024 (UTC)
 * Personally, I generally try not to mark as "minor" anything that other editors might plausibly think changes the content/meaning of the article. Things that I mark as minor edits include stuff like changing $$\frac12$$ to $$\tfrac12,$$ switching between "citation" and "cite X" templates to make the page style internally consistent, adding "clear" templates to fix layout bugs, switching hyphens to en dashes, adding wikilinks to jargon words, etc. The other type I often mark minor is follow-ups to a non-minor edit I made just before, e.g. rephrasing something that I decided, when rereading after committing the change, didn't quite match my intention. Other people might use the feature differently though. –jacobolus (t) 00:51, 16 June 2024 (UTC)
 * This sounds about right to me. I was thinking that the tweaks made here were not altering the meaning of the piece, but clearly opinions differ!  Chumpih  t 00:56, 16 June 2024 (UTC)

Bad phrasing in lede: 'inputs to the same inputs'
Here the troublesome clause is 'fed as inputs to the same inputs', which is not easy to parse. Perhaps a better form would be Sometimes a particular system is represented by a linear transformation whose outputs are fed back as inputs to the transformation. In addition, the phrasing 'In particular, it is often the case that' is perhaps better represented as 'sometimes'. Chumpih t 23:45, 15 June 2024 (UTC)


 * You are right that "fed as inputs to the same inputs" is awkward and could be rewritten in a better way, but changing the rest of the meaning of the sentence around it is unhelpful in a supposed copyedit. Including the word "feedback" separately, as was done before, is helpful because the word has developed to be somewhat independent of its origin as a phrase, so "fed back", even if wikilinked, is now relatively uncommon and unfamiliar and more likely to be confusing by itself. "In particular" and "sometimes" do not mean the same thing. The point of the former is emphasis. –jacobolus (t) 23:51, 15 June 2024 (UTC)
 * I take your point re. 'fed back' and 'feedback'.  But reducing verbiage is worthy, no?  What is the difference between "in particular, it is often the case that" and "sometimes" (apart from brevity)?  Chumpih  t 00:08, 16 June 2024 (UTC)
 * You have to read sentences in context. We have "The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in [applications]. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed ....". This sentence is awkward and could certainly be improved, but the point of this phrasing is to especially emphasize the case of systems modeled as linear transformations + feedback, because in these cases the largest eigenvalue and its associated eigenvector play a special role.
 * You changed that to "Sometimes a particular system is represented by a linear transformation whose outputs are fed ...". Changing from particular applications to particular systems is an unrelated use of the word "particular"; both removing the first particular and adding the second particular change the meaning and focus of the sentence. We're no longer calling out a specific special class of applications, or referring back to the previous sentence at all. "Sometimes" makes a clean break, not tightly tied to anything prior, and it's not at all clear what the new "particular" is even for. Readers can still infer from reading the rest of the paragraph that there's a relationship between these sentences, but that inference is not reinforced by the language. –jacobolus (t) 00:30, 16 June 2024 (UTC)
 * Fair enough, so how about ...[applications]. In such cases, models for the systems can include linear transformations that feed back their outputs to their inputs. In such representations, the largest eigenvalue is of use because it indicates the long-term behavior of the system, after many applications of the linear transformation, and the associated eigenvector is the steady state of the system. . Note that the model doesn't typically  "govern" a system, though i can represent or indicate what is happening.  Chumpih  t 00:52, 16 June 2024 (UTC)
 * As an aside, this part, while appropriate to a hypothetical nice article about eigenvalues and eigenvectos, is a bit out of place in this article as it stands, since we don't follow up by discussing it in more detail later. This makes it a bit of a bait-and-switch for readers. Ideally someone would come write a clear section of this article about systems modeled this way. –jacobolus (t) 00:35, 16 June 2024 (UTC)
 * Agree, that would be a great thing to see. Chumpih  t 00:38, 16 June 2024 (UTC)
 * @Tito Omburo originally wrote this bit in special:diff/1185535612. Maybe they have some ideas for cleaning up the phrasing a bit. –jacobolus (t) 23:53, 15 June 2024 (UTC)

Jorge Stolfi's version of the lead
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 or characteristic vector of a linear transformation $T$ is a non-zero vector $v$ that is only multiplied by some scalar factor $\lambda$ when transformed by $T$; that is $T(v)=\lambda v$. The scalar $\lambda$ is the corresponding eigenvalue or characteristic value associated with $v$.

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

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.

Any linear transformation of a space of dimension $d$ is completely defined by a set of $d$ 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 $d$-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 $$\mathbb{R}^2$$. The matrix has only integer entries (-1, 0,+1) but the eigenvalues are complex, $$\lambda_1 = +\mathbf{i}$$ and $$\lambda_2 = -\mathbf{i}$$.  The eigenvector of the first one is $$(1 + \mathbf{i}, 1 - \mathbf{i})$$, which is changed to $$(-1 + \mathbf{i}, 1 + \mathbf{i})$$. 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 $$d$$ mutually orthogonal eigenvectors", not "by the $$d$$ 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)

Missing terms
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)