Talk:Eigenvalues and eigenvectors

Latest comment: 3 months ago by Chumpih in topic Short description
Former featured articleEigenvalues and eigenvectors is a former featured article. Please see the links under Article milestones below for its original nomination page (for older articles, check the nomination archive) and why it was removed.
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Not too technical edit

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)Reply

Maybe, the less technically prepared get a less steep introduction by replacing the current first sentence with something like

Eigenvalues and eigenvectors belong in a characteristic way to a linear transformation, as dealt with in linear algebra. The transformation of an eigenvector results in just scaling it by the factor given by the eigenvalue belonging to this eigenvector. More formally ...

Just a suggestion. Purgy (talk) 08:58, 9 February 2019 (UTC)Reply
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).Reply
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)Reply
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)Reply

Historical origin of the use of lambda for eigenvalues? edit

My guess, it is from the early works of linear algebra and eigenvalues and eigenvectors arising from analyzing wave equations, where lambda would be used for wavelength, and different modes (eigenvectors) would correspond to various special solutions that can be linearly combined? Then set in stone when essentially same was done to Schrodinger's equation in Hamilton formulation of QM. Unfortunatly it is hard to find sources where the lambda symbol become popular for use for eigenvalues and what is the real origin of this popularity. 2A02:168:2000:5B:94CB:836:78C3:226E (talk) 12:17, 24 June 2020 (UTC)Reply

Lead is now ineffective, and possibly wrong edit

"an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes by a scalar factor when that linear transformation is applied to it. "

This fails to make clear that the salient feature of an eigenvector is that it is a vector in the direction in which the linear transformation applies no rotation. As it stands:

  1. the description is incorrect in that it doesn't exclude all the directions in which the linear transformation applies a scalar factor and a rotation.
  2. It does rule out genuine eigenvectors whose eigenvalue happens to be one.

I suggest some rewording that eliminates these incorrect aspects, and makes clear that eigenvector is about the direction of non-rotation, rather than whether or not there is scaling. Gwideman (talk) 14:11, 22 February 2021 (UTC)Reply

I don't see anything wrong with the definition above. In other directions a linear transformation need not be a rotation; it could, for example, be a sheer. The definition need not exclude other directions; the definition is about what happens to an eigenvector—not what happens to other vectors. It does not rule out eigenvalues of one; one is a valid eigenvalue and is encompassed by the definition above.—Anita5192 (talk) 17:03, 22 February 2021 (UTC)Reply
From the definition section:
"If T is a linear transformation from a vector space V over a field F into itself and v is a nonzero vector in V, then v is an eigenvector of T if T(v) is a scalar multiple of v. This can be written as
 
where λ is a scalar in F, known as the eigenvalue "
This is not the same as "changes by a scalar factor". It is the same as "changes only by a scalar factor, or remains unchanged".
To answer your points:
"the definition is about what happens to an eigenvector—not what happens to other vectors." Of course it's also about other vectors! We're trying to state criteria by which all those other vectors fail to qualify as eigenvectors.
"In other directions a linear transformation need not be a rotation; it could, for example, be a sheer." Shear describes the transformation of the plane (for 2D), not the transformation of an individual vector. When a shear is applied, most vectors rotate. Eigenvector identifies the ones that do not. There is an excellent visualization on YouTube channel 3Blue1Brown titled "Eigenvectors and eigenvalues | Essence of linear algebra, chapter 14" starting at 2:59, and at 3:12 "any other vector is going to get rotated". (Sorry, Wikipedia blocked the URL.)
"It does not rule out eigenvalues of one". The word changes rules out the scalar being 1. Gwideman (talk) 11:09, 1 March 2021 (UTC)Reply
I think the fact that the scalar could be one is a moot point. That is, it could be argued semantically that "changing" by a factor of one is not really "changing." Nonetheless I have reworded the lead slightly to clarify this.—Anita5192 (talk) 17:17, 1 March 2021 (UTC)Reply

Suggested re-ordering edit

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)Reply

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)Reply

Eigenvalues and the characteristic polynomial edit

The characteristic polynomial will only be monic if using the def

 

otherwise the def needs to be

 

where A is an nxn matrix.

See https://en.wikipedia.org/wiki/Characteristic_polynomial

Therefore I have changed the definitions in that section.

In my opinion,   should be used in all of this article. For finding eigenvalues it does not matter, but it will in other cases. And I believe it is better to have correct form from the start when beginning maths.

Edit: At least as far as I know, there is no case where the   would be a preferred, except for not having to do as many minus signs in ones equations :)

Mudthomas (talk) 13:09, 25 January 2022 (UTC)Reply

I reverted your change of order per Wikipedia:BRD. There is no need for the characteristic equation to be monic. Most textbooks use  .—Anita5192 (talk) 16:23, 25 January 2022 (UTC)Reply


Quoting the article on the characteristic polynomial, linked from the relevant section: "The characteristic polynomial   of a   matrix is monic (its leading coefficient is  ) and its degree is  ."
Edit: While I do not disagree that most textbook use  , I do not believe that they should :) -Mudthomas (talk) 19:45, 25 January 2022 (UTC)Reply
Quoting Wikipedia:When to use or avoid "other stuff exists" arguments, "In Wikipedia discussions, editors point to similarities across the project as reasons to keep, delete, or create a particular type of content, article or policy. These 'other stuff exists' arguments can be valid or invalid." Although the Characteristic polynomial article indicates that the polynomial is monic, I have never seen this in a reputable source.—Anita5192 (talk) 21:30, 25 January 2022 (UTC)Reply
I'm pretty sure I remember it from my courses in both ODE and Numerical linear algebra, but I wouldn't bet my life on it. I'll be back if I find corroborating sources! Mudthomas (talk) 21:39, 25 January 2022 (UTC)Reply
From [Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra, SIAM, Philadelphia, ISBN 0-89871-361-7, page 183:]
"The characteristic polynomial of Am × m, denoted by   or simply  , is the degree   polynomial defined by   Thanks to the placement of the minus sign,   is monic: the coefficient of its degree   term is 1. "
Furthermore the article on Characteristic polynomial cites the source [Steven Roman (1992). Advanced linear algebra (2 ed.). Springer. p. 137. ISBN 3540978372.] while the section here has NO source, reputable or otherwise. -Mudthomas (talk) 08:36, 26 January 2022 (UTC)Reply

Zero vector as an eigenvector removed. edit

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 (talkcontribs) 15:56, 10 February 2022 (UTC)Reply

Equation numbers are difficult to find. edit

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)Reply

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)Reply

Notation for determinant edit

Some sections in the article use bars for the determinant, even when applied to a symbolic expression, i.e.

  rather than  .

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:

 .

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

  FixedAnita5192 (talk) 15:11, 21 August 2023 (UTC)Reply

Short description edit

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)Reply

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)Reply
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)Reply
Vectors only scaled by linear transformations (45 characters) 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)Reply
This version is confusing to the point it's unhelpful, IMO. –jacobolus (t) 20:12, 16 November 2023 (UTC)Reply
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)Reply
As an aside, could you please skip past the revert war stage in similar future cases? –jacobolus (t) 19:44, 16 November 2023 (UTC)Reply
The current Short description is way too long — Vectors whose direction is fixed by a linear map, and the corresponding scalars (79 characters) — about twice as long as it should be. Please suggest sensible alternatives — GhostInTheMachine talk to me 19:50, 16 November 2023 (UTC)Reply
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)Reply
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)Reply
Short descriptions are not definitions. Johnjbarton (talk) 21:19, 16 November 2023 (UTC)Reply
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'easter (talk) 21:14, 16 November 2023 (UTC)Reply
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)Reply
Short descriptions do not need to be unique. They need to be short! Johnjbarton (talk) 21:19, 16 November 2023 (UTC)Reply
(edit conflict) So? We're not trying to fit a whole definition into it. XOR'easter (talk) 21:20, 16 November 2023 (UTC)Reply
Something like Mathematical concept linking vectors and matrices is more in the spirit, IMHO. Chumpih t 22:41, 16 November 2023 (UTC)Reply
This one seems nonsensical to me. Eigenvectors don't "link" anything. –jacobolus (t) 23:15, 16 November 2023 (UTC)Reply
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)Reply
Why not just say Mathematical concept, then? The title of the article itself already has vector in it, after all. XOR'easter (talk) 02:26, 17 November 2023 (UTC)Reply
Not unreasonable, but perhaps a little more required... Chumpih t 09:56, 17 November 2023 (UTC)Reply
I don't get it. What does linking vectors and matrices mean? XOR'easter (talk) 02:24, 17 November 2023 (UTC)Reply
perhaps Mathematical concepts involving vectors and matrices to then? Chumpih t 10:11, 17 November 2023 (UTC)Reply
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)Reply
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)Reply

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)Reply

Also a fine choice. Johnjbarton (talk) 15:37, 18 November 2023 (UTC)Reply
OK – No objections or further suggestions — GhostInTheMachine talk to me 20:28, 23 November 2023 (UTC)Reply
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)Reply
per Anita5192's suggestion, Concepts from linear algebra sounds OK to me. Chumpih t 22:47, 23 November 2023 (UTC)Reply

Not all vectors have direction edit

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)Reply

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)Reply
"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)Reply
"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)Reply
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)Reply