CourantFischer

Courant–Fischer refers to a fundamental variational principle in linear algebra and spectral theory that characterizes the eigenvalues of Hermitian (or real symmetric) matrices, and extends to self-adjoint operators on Hilbert spaces. Named after Richard Courant and Ernst Fischer, the theorem provides a way to understand eigenvalues through optimization of the Rayleigh quotient.

For a real symmetric or complex Hermitian n×n matrix A with eigenvalues λ1 ≤ λ2 ≤ … ≤ λn and associated orthonormal eigenvectors, the k-th eigenvalue can be expressed in two equivalent extremal forms. Let R(x) = (x^T A x)/(x^T x) denote the Rayleigh quotient. Then:

- λk = min over all k-dimensional subspaces S of R^n of the maximum of R(x) for x ≠ 0 in S.

- Equivalently, λk = max over all (n−k+1)-dimensional subspaces T of R^n of the minimum of R(x) for x ≠ 0 in T.

In component form, these yield the common two equivalent statements:

λk = min_{dim S = k} max_{x ≠ 0, x ∈ S} (x^T A x)/(x^T x)

and

λk = max_{dim T = n−k+1} min_{x ≠ 0, x ∈ T} (x^T A x)/(x^T x).

The Courant–Fischer principle also applies to compact self-adjoint operators on Hilbert spaces, giving a powerful framework for variationally characterizing spectra. It underpins numerical methods such as the Rayleigh–Ritz procedure and algorithms like the Lanczos method, and it is central to eigenvalue estimation and interlacing results.