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