Talk:Eigenvalues and eigenvectors

Colloquial definition in lead

The recently modified interpretation of the definition in the lead, "In the graphic setting of real vector spaces the direction of an eigenvector does not change, or is exactly reversed, under this transformation, just its length may be arbitrarily affected" is improper for several reasons. No matter how this is reworded, it will still have problems. I recommend removing it altogether. The first sentence, "In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by an overall scale when that linear transformation is applied to it" is enough to give the reader a feel for what an eigenvector is, without any inconsistencies, before reading the technical definition.—Anita5192 (talk) 16:52, 16 October 2017 (UTC)

The two edits before mine triggered my action: I felt compliant with the perception that scaling by (-1) does not leave the direction of a vector unchanged, and I wanted to add a visually supporting formulation to the formally fully correct version of scaling the vector. Not sure about the effects of my newbie support, I added, fully intentionally, a "?" to my efforts. Since there are recommendations of removal, I will undo my edit. Perhaps someone finds a better way to support the elementary graphic geometric aspects of real EVs. Purgy (talk) 17:27, 16 October 2017 (UTC)

I was not faulting you. I think all the recent edits were performed in good faith. However, regardless of everyone's good intentions, I think the first and third sentence are enough. I don't believe we need to refer to specific vector spaces to give an intuitive feel to something very general. Regards.—Anita5192 (talk) 17:40, 16 October 2017 (UTC)

I think something like this would work better as an image caption, much like the image in the overview section (although I also think a better illustration should be found for this purpose). But, as text, "graphic setting of real vector spaces" strikes me as likely to be confusing to the target audience. Sławomir Biały (talk) 22:49, 16 October 2017 (UTC)

I never felt faulted in the slightest way, and I certainly see the confusing aspects in my weak formulations, but I have no grasp on their improper-ness, and I still think that eliminating the previous remarks about direction (which still appear later on) makes the learning curve required for the lede steeper. Maybe an animation would be best. Purgy (talk) 07:42, 17 October 2017 (UTC)

What I thought were improper were several concepts either not yet defined, or not well defined, e.g., 1. graphic setting, 2. direction, 3. reversed, 4. length. I also thought it unrealistic to restrict the topic to real vector spaces. E.g., in Z₂, the only vector that could be an eigenvector is 1, and the only scalar multiples of 1 are 1 and 0. The terms direction and reverse are only trivially represented, at best.—Anita5192 (talk) 17:52, 17 October 2017 (UTC)

Adding Application to maxima-minima of multivariable functions

Eigenvalues are used in determining if a point is a local maximum , minimum or saddle point by calculating the eigenvalues of the Hessian Matrix the full article on it is Second partial derivative test. — Preceding unsigned comment added by Loneather (talk • contribs) 10:58, 6 December 2017 (UTC)

Etymology

The article claims that the German prefix "eigen-" means "proper" or "characteristic". I don't want to simply edit that statement, because it is linked to a source, but actually the main meaning of "eigen" is "own" (as in "my own", not as a verb): "mein eigenes Haus" = "my own house". So "Eigenwert" (eigenvalue) means something like "its very own value". Unless I'm missing a special meaning of "proper" (I'm German), this translation appears inappropriate to me. "Characteristic" fits better, but the main meaning "own" should be mentioned first in my opinion. 217.248.11.10 (talk) 21:24, 15 March 2018 (UTC)

The source cited reads:

eigen, adj. (Dat.) proper, inherent; own, individual, special; specific, peculiar, characteristic; spontaneous; nice, delicate, particular, exact; odd, strange, curious; ticklish;…

Eigenvectors are also called characteristic vectors in some textbooks. The "own" meaning is not the most relevant here.—Anita5192 (talk) 21:52, 15 March 2018 (UTC)

If you look beyond an English German-dictionary (Grimm or Adelung) you will find the Greek root ἔχειν, confirming the meanings of "property" and "ownership" as the core meaning of "eigen", used as a prefix, as adjective, or even as verb ("eignen") in a field with these notions as its center. Former ages, where personal property determined the perceived personality to a greater extent, already coined the view that "property, one owns" makes up (to a good deal) the "character" of a person. As I perceive it, the translations of "eigen" to "own", "characteristic", "specific", ... immediately hit the spot. Maybe, the intended meaning of "proper" in this context, prefixed to the noun ("proper value") is less immediate to a non-native speaker, compared to the postfixed use ("value proper"). Other translations, given in the source, result from a (factual) slight shift in meaning to the pejorative side ("peculiar", "strange" (="curious"), ..., but still "characteristic"), others are -say- rare, if not curious ("spontaneous", "exact", "ticklish").

As a natively German-speaker (I'm Austrian) I would -unauthorized, but spontaneously- prefer to use "own" wrt material goods, but abstract conceptions were "proper" to me. Honestly, I do think that linear maps "own" their "eigen"values, in the same sense as I "own" (without any rights) my mental conceptions of meanings. So maybe, changing the order of "own" and "proper" is merited, or it is not. :D Cheers, Purgy (talk) 08:28, 16 March 2018 (UTC)

The source cited supports the possible meanings of the prefix, but not the specific choices of "proper" and "characteristic." The editor who first inserted this evidently left no source or explanation for his or her choice of meanings. Most textbooks refer to "eigenvectors" as "characteristic vectors," but do not use other meanings. I would like to know the history of the term "eigenvector" and why the prefix "eigen–" was chosen, but none of my sources address this. The German article at [[1]] indicates that "eigen–" means "characteristic quantities," and dates to a publication by David Hilbert in 1904. Perhaps we should remove the prefix "proper" from the article.—Anita5192 (talk) 18:08, 16 March 2018 (UTC)

Not 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

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)

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)

Historical origin of the use of lambda for eigenvalues?

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)

Lead is now ineffective, and possibly wrong

"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)

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)

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

$T(\mathbf {v} )=\lambda \mathbf {v} ,$

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)

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)

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)

Eigenvalues and the characteristic polynomial

The characteristic polynomial will only be monic if using the def

$p_{A}(\lambda )=det(\lambda I-A)$

otherwise the def needs to be

$(-1)^{n}p_{A}(\lambda )=det(A-\lambda I)$

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, $\lambda I-A$ 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 $A-\lambda I$ 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)

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

Quoting the article on the characteristic polynomial, linked from the relevant section: "The characteristic polynomial $p_{A}(t)$ of a $n\times n$ matrix is monic (its leading coefficient is $1$) and its degree is $n$."

Edit: While I do not disagree that most textbook use $(A-\lambda I)$, I do not believe that they should :) -Mudthomas (talk) 19:45, 25 January 2022 (UTC)

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)

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)

From [Lloyd N. Trefethen and David Bau III, Numerical Linear Algebra, SIAM, Philadelphia, ISBN 0-89871-361-7, page 183:]

"The characteristic polynomial of A ∈ ℂ^{m × m}, denoted by $p_{A}$ or simply $p$, is the degree $m$ polynomial defined by $p_{A}(z)=\det(zI-A).$ Thanks to the placement of the minus sign, $p$ is monic: the coefficient of its degree $m$ 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)

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)

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)