Talk:Eigenvalues and eigenvectors

Now that we have all this information in one place we need to figure out the correct structure. --MarSch 3 July 2005 12:20 (UTC)

Inifite-dimensional spaces

The concept of eigenvectors can be extended to linear operators acting on infinite-dimensional Hilbert spaces or Banach spaces.

Why is this called an extension? It looks like a straightforward application of the definition. Also, is there really a restriction to Banach spaces in infinite-dimensional cases? In any case, "Hilbert spaces or Banach spaces" is redundant since all Hilbert spaces are Banach spaces. Josh Cherry 3 July 2005 17:17 (UTC)

You're right, the definition of eigenvectors and -values does not depend on the dimension. But the spectrum of an operator is an extension of eigenvalues.

Power method

For the matrix having 2 on the diagonal and 0 off diagonal, the power method as explained in this article will fail to converge to any eigenvector, since applying the matrix is the same as multiplying by two, and this is done an infinite number of times. Am I getting something wrong? Oleg Alexandrov 4 July 2005 00:06 (UTC)

Uhm, yes, well, I have to admit that "converge" is used in a very loose sense here. I tried to amend the text. It is still work in progress anyway. Somewhere on Wikipedia it should also be mentioned that the method fails with matrices like

{\begin{bmatrix}0&1\\1&0\end{bmatrix}},

which has two eigenvalues of equal magnitude. Sigh, so much to do ... -- Jitse Niesen (talk) 4 July 2005 00:37 (UTC)

The claim that insolvability of the quintic by radicals implies no algorithm to solve polynomials in general is wrong. I have rephrased that section. Stevelinton 09:56, 29 August 2005 (UTC)[reply]

Physical significance of eigenvectors

I find the article to be fairly accessible to those who are not math gurus but too much emphasis in the article is on the abtruse language generally employed by mathematicians to describe what I colloquially term "eigenstuff". What may help is if someone adds some of the physical significances of eigenvalues and eigenvectors, such as:

The moment of inertia tensor always produces orthogonal eigenvectors which correspond to principal rotational axes about which the equations of rotational motion become very simple. The eigenvalues are the scale factors of the moment of inertia equation in such a case.
A matrix which describes an approximation of a conjugated pi system (The Hueckel Approximation, if anyone is curious) often appears as a determinant, and setting it equal to zero yields the eigenvalues (allowed energies) and the eigenvectors (normalized but not necessarily mutually orthogonal in whatever vector space it spans, not that chemists care about the vector space involved) yield the contribution of each atom in the pi system to the associated molecular orbital represented by the eigenvalue.
In heat transfer the eigenvectors of a matrix can be used to show the direction of heat flow. I am not entirely sure in this case what the eigenvalues represent. My thermodynamics didn't extend to matrix versions of the Fourier equation, so... (???)
Someone mentioned stress-strain systems where the eigenvectors represent the direction of either greatest or least strain or stress, depending, IIRC, on the way the matrix is set-up. Again, not having done engineering in detail I can't be more specific.
More generally someone mentioned to me once that in spectroscopy you can set up a matrix which represents the molecule under study, and in Infrared in particular, because it effectively acts as a series of coupled springs which vibrate at discrete energies, the eigenvalues here give the energies (wavenumbers) for the peaks and the eigenvectors tell what kind of motion is happening. Again, without quantum spectroscopy in detail I can't be more specific than this.

I would like to encourage Wikipedians in general, if they are editing math-specific entries, to try and relate the significance, either mathematically, or physically, of the concept or method in less abtruse language than is typical in linear algebra texts. :)

--142.58.101.46 4 July 2005 16:23 (UTC)

Be bold and start editing. I agree that sometimes mathematicians push abstraction to the point where nobody else understands what is going on. Oleg Alexandrov 5 July 2005 02:36 (UTC)

I think the mathematics and the physics belong in seperate articles. The scope of this one is just too large. Perhaps the maths could go in "Spectrum (Linear Algebra)", the physics applications in "Eigenvalues and Eigenvectors in Physics", and other applications in "Applications of eigenspaces outside physics"? From a mathmatical point of view, the chain of associations sort of goes Linear Operators (-> Spectra) -> Differential Operators -> Differential Equations (-> Solutions) -> Applications, while in physics it goes Eigenstuff -> {Energy states, Principal axes, Modes, Observables, ...}. I think that combining these into one article will make it hard for physics or mathematics readers (or both) to navigate and stay motivated.

Also, the categories and names in physics and mathematics are quite different. Physicists think of rotations in R^3 and observables in Hilbert space as very different things, but to mathematicians they're all linear operators, though the observables are a special type called Hermitian. R. E. S. Polkinghorne

Eigenfunctions belong here too?

Yes? No? --HappyCamper 06:12, 11 August 2005 (UTC)[reply]

Singular Values ?

A reference to singular values and singular value decomposition could be of interest too ? --Raistlin 16:16, 13 August 2005 (UTC)[reply]

I agree that it looks bad in the TOC, but it seems wrong that we should be compromising the article in order to improve the æsthetics of a standardised TOC. Has this problem been solved satisfactorily elsewhere on WP? (The obvious suggestion would be an option for subsubsubsections (or whatever) not to be mentioned in the TOC at all.) —Blotwell 09:26, 18 August 2005 (UTC)[reply]

Projections

projection onto a line: vectors on the line are eigenvectors with eigenvalue 1 and vectors perpendicular to the line are eigenvectors with eigenvalue 0.

Isn't that true only of orthogonal projections? Consider

$A={\begin{bmatrix}0&0\\1&1\end{bmatrix}}.$

Doesn't that qualify as a projection onto a line? Yet it has no eigenvectors perpendicular to the line. Josh Cherry 14:35, 20 August 2005 (UTC)[reply]

You are right, I fixed it.--Patrick 09:16, 21 August 2005 (UTC)[reply]

intuitive definition

the definition as given is, i think, unnecessarily hard to visualize. i would say that an eigenvector is any unit vector which is transformed to a multiple of itself under a linear transformation; the corresponding eigenvalue is the value of the multiplier. that is really clear and intuitive. the fact that a basis can be formed of eigenvectors, and that the transformation matrix is diagonal in that basis, is a derived property. starting off with diagonal matrices and basis vectors is unnecessarily complicated. ObsidianOrder 22:49, 21 August 2005 (UTC)[reply]

P.S. that way, eigenvalues and eigenvectors in infinite-dimensional spaces are not an "extension" either, they just follow from the exact same definition. ObsidianOrder

I agree. Josh Cherry 01:26, 22 August 2005 (UTC)[reply]

Yes.. the idea of forming a basis of eigenvectors does NOT belong in the intro. Here's an idea -- perhaps there should not be an intro? Perhaps the first part of the definition, the non-formal, geometrical part that you describe, can serve as the intro? Or.. if the format that needs to be followed is that there must be an intro, then nevertheless it should go from less technical to more technical, so the intro should have your non-formal geometric definition and in the definition section we can put the formal definition. The whole thing about diagonalizing a matrix should be mentioned, but LATER. Kier07 03:00, 22 August 2005 (UTC)[reply]

I think the most intuitive idea of eigenvalues and eigenvectors comes from principal component analysis, and is reflected in the main applications. You want to express the transformation in terms of invariant axes, which correspond to eigenvectors, and how these contract or expand corresponds to the eigenvalues. --JahJah 02:56, 22 August 2005 (UTC)[reply]

Perhaps. Here's another idea... would it be possible to have a visual for the intro, instead of this verbal nonsense? We could give a linear transformation from R^2 --> R^2, and show in a picture what this does to a sampling of vectors -- including invariant axes. Kier07 20:35, 22 August 2005 (UTC)[reply]

At the moment we're giving the mathematical coördinate-free definition in terms of linear transformations, which obscures the connection with many of the applications. Often matrices don't represent transformations, but bilinear forms—this is when they have two covariant/two contravariant indices rather than one of each, if you're into tensors—and we should point out that eigenvectors are physically meaningful in this case too. This is where all the applications that give you symmetric matrices come from. (There is a relation between transformations and bilinear forms, of course, but it wouldn't be helpful to go into it in the introduction.)

Following on from this, here's the idea I had been thinking about for an introductory visual: if you have a symmetric matrix and apply it to the unit sphere, then you get an ellipsoid whose principal axes have direction corresponding to the eigenvectors and magnitude corresponding to the eigenvalues. (Check: the product of the magnitudes is proportional to the volume, which is right because the product of eigenvalues is the determinant which is the volume scale factor.) This should be easy/fun/illuminating to illustrate. —Blotwell 05:50, 23 August 2005 (UTC)[reply]

motivation?

A little while ago I suggested a sentence in the introduction to give some motivation, something really simple to tell the reader why they should care. It was along the lines of:

eigenvectors/eigenvalues etc allow us to replace a complicated object (a linear transformation or matrix) with a simpler one (a scalar).

This simple idea, expressed in very plain english, explains why we care about eigenvalues/vectors, without confusing the uninitiated with terminology like bases, diagonal entries, etc. The sentence was deleted at some point, and I feel the omission with some pain. Does anyone else agree that such a sentence would be a Good Thing? Dmharvey File:User dmharvey sig.png Talk 02:30, 23 August 2005 (UTC)[reply]

Me, I agree. Though I'm no pedagogue. —Blotwell 05:52, 23 August 2005 (UTC)[reply]

Where eigenvectors come from

In the section on "Identifying eigenvectors", it was not immediately clear where the vector (1, 1, -1) came from. It turned out it was the solution for $(A-\lambda I)x=0$ , and this turned out to be pretty easy. But it would be nice to explicitly state where to find the "sets of simultaneous linear equations" to be solved.

WillWare 23:14, 25 August 2005 (UTC)[reply]

Improving further

I have done a lot of work with respect to the intro and definition. I think some of the things I wrote are not optimized and are redundant with infos somewhere else on the page. Now I am tired and would like to see other authors contribute to a better structured article. 131.220.68.177 09:31, 8 September 2005 (UTC)[reply]

nonsense

Do we want to make any mention here about the fact that many disciplines have started to use "eigenvalue" as a buzz word? I was talking to a philosophy professor recently, a man with numerous published books, and when he began talking about "eigenvalues" I asked him how he was using the term. He was completely taken aback, and finally admitted he had no idea what the term meant, but it was in common use in his field.

Please do, with examples. The abuse of "parameter" may finally be dying; let's kill this one now. Septentrionalis 02:11, 14 September 2005 (UTC)[reply]

You're kidding. Wow. I've never heard of arts-type disciplines using the term "eigenvalue". I'll have to ask the philosophy students I know if their professors do this. I think the overuse of the term may come from the fact that it shows up so much in many sciences (for valid reasons). --24.80.119.229 18:43, 18 September 2005 (UTC)[reply]

Eigenwert which is German for eigenvalue has a broader sense than in mathematical English. It can be also translated as distinct value see discussion in German at [1] because eigen is a very common prefix in German and can be used in a very broad sense. In that sense one could say The eigenvalue of the president's speech was more formal than objective. See for example Eigenwert des Ästhetischen at [2]. So sometimes, the term eigenvalue can be used in this sense for example:

Thus one can say that research and therefore science fulfills a function and thereby reproduces a stable eigenvalue of modem society. One cannot simply refrain from research without triggering catastrophic consequences -- catastrophe understood here as the reorientation towards other eigenvalues.[3]

I am no philosopher but as far as I understand this. It means that science is a value of a modern society which makes it clearly distinct from a more archaic society. Vb 09:58, 19 September 2005 (UTC)[reply]

The paragraph above is a translation into very unidiomatic, and almost unreadable, English; the whole text almost reads like a Babelfish exercise. The use of the ~~claque~~ calque "eigenvalue" in this context is simply an error, as is "reproduces". Eigenwert would have been acceptable, so would "distinctive value". In short, Vb has given a diagnosis, not a justification. Septentrionalis 17:28, 19 September 2005 (UTC)[reply]

I presume that you meant calque. Josh Cherry 00:06, 20 September 2005 (UTC)[reply]

I utterly agree this is a diagnosis not a justification. One needs more examples of this kind of usage to begin some article/section/disambig on this. Vb 81.209.204.158 16:32, 20 September 2005 (UTC)[reply]

a duplicate/related article

There's a recently spawned article Symbolic computation of matrix eigenvalues which I don't much like; it seems to be a spinoff of this project, maybe? It needs attention or cleanup or something. linas 01:01, 14 September 2005 (UTC)[reply]

Yes indeed. This info was previously within this article. I put it in a separate article because I thought this info maybe interesting for grad students fighting with matrix diagonalization but not for someone interested in the general topic of this article. I think this is the same for eigenvalue algorithm. Maybe the good idea would be to merge Symbolic computation of matrix eigenvalues with eigenvalue algorithm. Vb 10:13, 14 September 2005 (UTC)[reply]

Featured article?

I think now the article is getting quite good. However I don't think this is enough for getting featured. I think the following points have to be addressed before submitting for peer review:

Copy edit : I think the English is not the best. I am no native speaker and I cannot manage it. That's the reason why I put the copy-edit flag on the page.
Too technical : Some info on the page are too technical. In particular many properties are stated without explaining the reader why they are important and in which context they are used. In particular, the sections Decomposition theorem, Other theorems, Conjugate eigenvector, Generalized eigenvalue problem and Eigenvalues of a matrix with entries from a ring should be re-written to be more accessible to a broader audience.

I think much of the info here is good and interesting but sound a bit too much like a grad book for mathematicians. Vb 16:16, 21 September 2005 (UTC)[reply]

Problem with one of the examples

The following passage occurs as an explanatory example:

If one considers the transformation of the rope as time passes, its eigenvectors (or eigenfunctions if one assumes the rope is a continuous medium) are its standing waves well known to musicians, who label them as various notes. The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as the time evolves.

A string of random length stretched at random tension can produce a sound that corresponds to a particular frequency. But depending on various factors the string can perform a simple vibration in which the entire length of the string moves in the same direction at the same time, or it can produce more complex vibrations (like the one illustrated in the movie). The musical qualities of these various possible variations are emphasized in the reader's mind by the mention of musicians because a pure sine wave sound vibration is not musically beautiful. Not only will the reader's mind potentially be sidetracked by that line of thought, but it is not exactly true that a musician will label any frequency as a "note" -- particularly if the musician has perfect pitch and the frequency being prduced is somewhere midway between a A and an A flat. I don't want to change this passage without being aware of what the original writer was trying to accomplish. P0M 05:37, 23 September 2005 (UTC)[reply]

I am no musician. This example I wrote is a typical physicist's example which has not much to do with music. Here the objective was only to provide an example of eigenvector in an infinite dimensional space which could be understood by nonspecialists. If someone is able to make a better link with music, I would be very happy if he could do so. I would learn something! However I think it is important to keep in mind that one should not go into deep details here. This is not the place to present the whole theory of vibrating strings. Vb 07:51, 23 September 2005 (UTC)[reply]

If you weren't trying to say something tricky about higher harmonics, Martin guitar vs. Sears guitar, etc., then how about:

If one considers the transformation of the rope as time passes, its eigenvectors (or eigenfunctions if one assumes the rope is a continuous medium) are its standing waves -- the things that, mediated by the surrounding air, humans can experience as the twang of a bow string or the plink of a guitar . The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as the time evolves.

P0M 15:11, 23 September 2005 (UTC)[reply]

Yes that's true I had also higher harmonics in mind. However, I believe it is not worth to tell more than a line about it or maybe just a link. What you wrote is from my point of view well done. I would replace "the things that," by "which" but my English is for sure not as good as yours. My problem is also that I am a bit afraid of telling nonsense about music: I have no idea about Martin and Sears guitar. But if you do why not making a footnote about such details: they are for sure interesting (at least to me) even if they would disturb a bit from the main topic of this article. Vb 09:04, 24 September 2005 (UTC)[reply]

I made the basic change. If something were to be added maybe we could say something like, "Waves of various degrees of complexity may be present, analogous to anything from the dull plunk of a cigar box string instrument to a chord from a Stradivarius violin"? (Martin guitars are not quite that good, and certainly not that expensive anyway. Sears used to sell cheap guitars that hardly "sang" at all -- sorry, that is definitely POV. ;-) How do the higher harmonics get set up in quantum situations? P0M 01:48, 27 September 2005 (UTC)[reply]