the Courant-Fischer min-max principle
/ koo-RAHNT FISH-er /
The Rayleigh quotient already tells you the smallest eigenvalue is the minimum of the energy ratio over all admissible functions. But how do you reach the SECOND eigenvalue, or the hundredth, without first knowing all the ones below? The Courant-Fischer min-max principle is the answer: it characterises every eigenvalue, individually, as the solution of a nested minimisation-and-maximisation game — no knowledge of the lower eigenfunctions required.
For the Dirichlet Laplacian with Rayleigh quotient R[u] = (the integral of |grad u|^2) / (the integral of u^2), the n-th eigenvalue is lambda_n = the minimum over all n-dimensional subspaces S of admissible functions, of the maximum of R[u] over nonzero u in S. In words: look at every possible n-dimensional subspace, find the worst (largest) Rayleigh quotient inside each one, and then lambda_n is the best (smallest) such worst-case. There is a dual max-min form too. The minimising subspace is the span of the first n eigenfunctions, and on it the maximum is achieved by the n-th eigenfunction itself. For n = 1 the subspaces are just lines and the principle collapses back to the plain Rayleigh quotient minimisation.
The payoff is a powerful comparison tool. Because the characterisation involves no eigenfunctions — only subspaces and the energy ratio — you can bound eigenvalues by clever choices of trial subspaces. Shrink the domain and every eigenvalue goes up (a smaller drum rings higher); add a positive potential and every eigenvalue rises; this monotonicity is immediate from min-max and would be awkward to see otherwise. It underlies eigenvalue interlacing, the convergence of the Rayleigh-Ritz and Galerkin numerical methods (whose computed eigenvalues are always upper bounds), and Weyl's asymptotic law. It is the variational soul of the spectrum.
To bound lambda_2 of a membrane from above, pick any 2-dimensional trial subspace — say the span of two simple bump functions that vanish on the boundary. Compute the largest Rayleigh quotient over that pair; min-max guarantees lambda_2 is no larger than that number. No eigenfunction needs to be known, and a sharper pair of bumps gives a tighter bound. This is exactly how a finite-element code estimates eigenvalues from above.
Eigenvalues as a subspace game — choose a trial subspace, take the worst ratio inside, and you have an upper bound.
The min-max formula gives upper bounds when you plug in trial subspaces (the inner max over a too-small space can only be smaller than the true outer min). Getting matching LOWER bounds is genuinely harder and usually needs the dual max-min form or extra structure — so a single Rayleigh-Ritz computation pins eigenvalues only from above, never from below.