Talk:Courant minimax principle

Rewrite

This article does not seem to be accurate. The minimax princple is usually credited to Rayleigh and Ritz in numerous textbooks. The principle applies equally well in infinite dimensions, as is made clear in the book of Courant and Hilbert, where the very elementary proof (which does not use induction) is given. In the article no applications of the principle are given, e.g. to estimating the kth eigenvalue of a sum of self-adjoint matrices. There are a host of other such inequalities related to minimax, equally valid in infinite dimensions, which can be found in the book of Barry Simon on Trace Ideals. I do not believe that Courant was the first to prove the maximum-minimum principle. At some later date I might rewrite and rename this page (to "Minimax principle"). Mathsci (talk) 08:45, 19 April 2008 (UTC)

Regarding the naming, I think the author of this page was fine in his choice, and I wouldn't say that it's reason enough to "dispute the factual content" of the page. There is often no universal convention to naming. In my copy of Horn and Johnson, they call this the "Courant-Fischer" theorem, and they state it in full min(max(...)) and max(min(...)) terms, for the kth largest eigenvalue of a Hermitian matrix, and then remark that if k = n or k = 1, it reduces to what they call the Rayleigh-Ritz Theorem.

Franklin's "Matrix Theory" also calls this the "Courant minimax" theorem, and applies it to Hermitian matrices. Just like Horn and Johnson, they also refer to "Rayleigh's principle" as the case when k =1, i.e. lambda_1 = max <Hx,x> / <x,x> . My copy of Barry Simon's "Functional Analysis" doesn't refer to the theorem at all. So I can't say that I agree with Mathsci.

The article should be *expanded*, that's for certain. Change "real and symmetric" to Hermitian, and include, as Mathsci suggests, infinite dimensions. Also, instead of just the min(max()) formulation, include the max(min()) formulation as well. Lavaka (talk) 00:24, 17 May 2008 (UTC)

The minimax principal is used beyond elementary matrix theory, which is not its main application. For example it plays a fundamental role in estimating the eigenvalues of a Laplacian on a compact Riemannian manifold (see for example Chavel's book on Eigenvalues in Riemannian geometry). This is part of so-called "comparison geometry". It is discussed twice in Courant and Hilbert, the second infinite-dimensional application being the more significant, The main point was that for other mathematical articles on the WP, this was not a helpful article. In its initial state, it was not an appropriate encyclopedic reference for Spectral theory of ordinary differential equations. There, because of the state of the current article, I gave my own statement and proof. For the standard Sturm-Liouville problem on a compact interval, minimax is the only tool available for estimating eigenvalues since the problem can only be solved in closed form when the potential q(x) is a step function. Barry Simon did not write a book on functional analysis - do you mean Reed and Simon, Mathods of Mathematical Physics Vol I? The content of WP need not rely on which particluar textbooks WP editors possess. It is for example discussed at great length in Barry Simon's book on trace ideals. The whole theory of s-numbers is also developed in the book of Pietsch. It is also discussed at length (without proofs) in Connes' book on Noncommutative Geometry. This is a complaint about the poor content and unencyclopedic nature of the page. I don't think it's disputed that Rayleigh and Ritz preceded Courant and Hilbert in using calculus of variations to study eigenvalue problems (in the theory of sound). In my experience most mathematicians don't call this Courant's minimax principle, simply the minimax principle, so the title is misleading. It would be totally inapproriate to a longer article on the uses of minimax in higher mathematics with the current title. I don't see why "Minimax Principle" should cause any problems. In the section on applications to matrices, reference to Courant can be given: look on the Jahrbuch für Mathematik, the references in Courant and Hilbert or other books on Methods of Mathematical Physics. No particular conclusion can be drawn from its presence or absence from texts on functional analysis (e.g. Avner Friedman's Foundations of modern analysis or the first volume of Dieudonné's treatise on analysis).

It would certainly be useful to locate where the result was first established for second and higher eigenvalues. The matrix case is treated in Courant and Hilbert Vol I, pages 31-32, with an application on pages 33-34. The references on page 47 cite textbooks by Bocher, Kowalewski and Wintner and monographs/articles by Courant (Zur Theorie der Kleinen Schwingungen, 1929), Fischer and Hilbert (Foundations of the theory of linear integral equations). Please look at these references and the others that I mention before making assertions about history or content. Courant and Hilbert's classic book is probably one of the definitive sources. Whether subsequent mathematicians or numerical analysts have chosen to follow their treatment or not is a separate issue, irrelevant to WP. Mathsci (talk) 14:53, 17 May 2008 (UTC)

I don't really intend to get in an argument over this. Mathscinet quoted some books, so I just intended to do the same. The Simon book I was referring to is indeed "Methods of Mathematical Physics Vol I", and the title of it is "Functional Analysis". Authors are Michael Reed and Barry Simon. Not sure why Mathscinet brought that up. Anyhow, I don't want to argue about what the most important applications of the minimax principle are. I do agree that the article should be expanded. Lavaka (talk) 22:05, 19 May 2008 (UTC)

I brought this up because Barry Simon wrote a book (Trace ideals and their applications) where a large part of the subject is discussed in detail. It seems pointless referring to other books by Barry Simon which don't treat the subject. Have you looked at the classic textbook Courant and Hilbert, Volume I? It is unclear why you bother mentioning books that do not treat the subject: there you completely mystify me. As I said, I could not use the present article elsewhere on the WP. (I had a similar but more serious problem with Fredholm determinants, which I almost completely rewrote.) It was simpler at the time to include elsewhere the 2 simple sentence proof (which I have taught many times to undergraduates and graduates) rather than bother to improve this article. Just as a matter of interest, I mentioned "Courant's minimax principle" to a colleague on the Berkeley maths faculty and he had exactly the same reaction about the reference to Courant as I had. At some stage, when I have time, I might simply rewrite the article, using the standard sources I have already mentioned. It is not the name of this article so much as its content ( which is unencyclopedic) that I find unsatisfactory. The fact that I had to add one of the original principal sources (Courant & Hilbert) was not a good sign at all. Please go and read that reference before making further comments. Mathsci (talk) 06:27, 20 May 2008 (UTC)

Your arrogance and hostility is astounding, and your talk page further confirms this. I do not wish to discuss mathematics with you. Lavaka (talk) 15:05, 20 May 2008 (UTC)

Your personal attack has been noted. I have been advised by a fellow mathematical wikipedia editor here in Berkeley that your comments on this page should be ignored. (This editor should not be confused with Lord of the Winds, whose recent comments I removed from my talk page, although they will remain topical until 3.30pm on Friday, California time.) Mathsci (talk) 21:04, 21 May 2008 (UTC)

Rewrite Redux

Yes, this article should be rewritten. I will try to come back to it later. If Courant is the father of this principle for those working in matrix analysis, then its use in mathematical physics is probably first (e.g. Rayleigh, Ritz). The article needs to be revised to allow for the opertaors of mathematical physics e.g. unbounded semi-bounded self-adjoint operators. (Courant would probably be amused by this article....) Underlying the principle is the notion of eigenvalues of quadratic forms. Kato's classic text "Perturbation theory for linear operators" has a very nice treatment. More illuminating historical references might be found there too. Lost-n-translation (talk) 07:26, 5 November 2009 (UTC)

Proof, generalizations & applications

Absent any issues of whom the result should be named after, I think it would be beneficial to add a proof for the matrix case, a section on generalizations to operators in Hilbert spaces and another section with simple applications. The article on this very important topic should not be a stub. I'll start adding this in the coming days and welcome any comments and/or blind rage.

As for the naming, it might be helpful to rename it to just plain Min-max Principle and have "Courant Min-Max Principle" or "Rayleigh's Principle" all point to the same article. That way people who are searching for it but don't know it by one name or another will be more likely to find it. Compsonheir (talk) 07:54, 13 June 2011 (UTC)

Since I've been quoted in the comments (I came here because I was trying to google where Courant used min-max!), let me say the impression I always had is that Rayleigh and Ritz use max and min for the top and bottom eigenvalues and realized that other eigenvalues are saddle points of the quadratic function but did not have the min-max formula which is due to or at least pushed by Courant. In particular A<B imples all the eigenvalues of B are larger than those of A - this does not follow from Rayleigh-Ritz but does from min-max. All this said, this is an impression that I got from my teachers 40 years ago and is not serious history. By the way, while Reed-Simon Vol I probably doesn't mention min-max, I'm reasonably sure Reed-Simon Vol IV does heavily as well as my Trace Ideal book.

In any event I agree the sensible thing is to have a page called "Minmax Principle" with pointers for searches on Rayleigh-Ritz and Courant-Fisher and that the page should have both a proof for finite matrices and the application to monotonicity of eigenvalues. Absent real history, it should be mentioned that it associated witrh Rayleigh, Ritz, Courant and Fisher. -- Barry Simon Bsimonca (talk) 12:04, 23 June 2011 (UTC)

PS: There already is an article on min-max. This article should be dropped with some kind of merging to the existing min-max. Bsimonca (talk) 12:07, 23 June 2011 (UTC)

The WP organization of this material is quite chaotic! It is treated in multiple different articles, apparently written by people unawere of the others. There are articles Rayleigh quotient and Min-max theorem , the last one seem to be the most complete and probably all the material should o in that article, and the two other ones being redirects. Kjetil B Halvorsen 21:18, 11 February 2014 (UTC) — Preceding unsigned comment added by Kjetil1001 (talk • contribs)