The maximum eigenvalue of a real symmetric (or complex Hermitian) matrix is given as the maximum of the associated the quadratic form:

$$ \lambda_{\rm max}(A) = \max_{\|x\| = 1} x^*Ax. \tag{1} $$

This result is often attributed to Rayleigh and Ritz (for example, in Fuzhen Zhang's Matrix Theory).

But was this result actually originally proven by either Rayleigh or Ritz? In Stewart and Sun's Matrix Perturbation Theory, the authors write

Both Rayleigh and Ritz were concerned with approximating the eigenvalues of an infinite operator by replacing it by a matrix eigenvalue problem... Neither gave formal justification for his method.

Stewart and Sun referenced work by Rayleigh in 1899 and Ritz in 1909. Fischer's theorem, which contains the "Rayleigh–Ritz theorem" (1) as a special case, was proven in 1905, four years earlier than the work of Ritz cited in Stewart and Sun.

There are two separate but related ideas attributed to Rayleigh and Ritz:

  1. The numerical method of approximating extremal eigenvalues of a matrix/operator by maximizing the quadratic form (or Rayleigh quotient) over vectors from a subspace.
  2. The variational principle for extremal eigenvalues. That is, the truth of equation (1) as a theorem of mathematics.

Based on the account in Stewart and Sun, it seems like that Rayleigh–Ritz are correctly attributed for developing idea 1, but idea 2 seems more properly to be attributed to Fischer, at least as a rigorous mathematical theorem. At the very least, it seems Ritz has no claim to originating equation (1) as their published work on the matter followed that of Fischer.

In summary, my question is the following:

Who was the first to establish equation (1), either informally or formally?

  • 1
    $\begingroup$ I think it is more typical to attribute to them the Rayleigh-Ritz method. The theorem, or rather its generalization that expresses all eigenvalues, Wikipedia, for example, calls Courant minimax principle (real case) and Courant–Fischer–Weyl min-max principle (Hermitian case). Although, considering that Courant's first publication is from 1910, Fischer apparently proved it before him. $\endgroup$
    – Conifold
    Sep 9, 2022 at 21:42
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    $\begingroup$ According to Parlett, The Rayleigh Quotient Iteration, Rayleigh considered this (in terms of quotients named after him) since 1870s in the context of acoustics, and was, apparently, aware of the result, although I am not sure if he proved it. The first edition of his Theory of Sound is from 1877. $\endgroup$
    – Conifold
    Sep 9, 2022 at 21:55
  • $\begingroup$ Thanks for these very helpful comments @Conifold. I agree it is ubiquitous to attribute to them the Rayleigh–Ritz method. I assume their association with the method is related to why some authors (e.g., Zhang as linked in my most) also attach their name to the theorem/principle as well (which as your suggest is a special case of Fischer's theorem). The references you found definitely provide strong evidence that Rayleigh was aware of the result, and if you attach it to Rayleigh might as well add Ritz I guess. Thanks for your help! $\endgroup$
    – eepperly16
    Sep 10, 2022 at 15:55
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    $\begingroup$ Regarding the assertion that Rayleigh and Ritz are correctly acknowledged for having conceived the numerical method for approximating eigenvalues ("... it seems like that Rayleigh–Ritz are correctly attributed for developing idea 1 ..."), the paper "From Euler, Ritz, and Galerkin to Modern Computing" (SIAM Review, Vol. 54, No. 4) suggests that the inclusion of Rayleigh's name is historically inapprorpriate (specifically, see page 655, journal pagination). $\endgroup$ Sep 11, 2022 at 6:47
  • $\begingroup$ @MarkYasuda Thanks for the correction. The SIREV article (and the article it cites) make a compelling case that Ritz substantially improved on Rayleigh's original method. Based on my current understanding, it doesn't seem inappropriate to attribute the method to both Rayleigh and Ritz (Ritz's method does build conceptually on Rayleigh's approach) or just Ritz alone (for being substantially different and better than Rayleigh's), but not just to Rayleigh $\endgroup$
    – eepperly16
    Sep 11, 2022 at 18:37


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