Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices
Suk-Geun Hwang · American Mathematical Monthly · 2004
Hermitian matrices have real eigenvalues. The Cauchy interlace theorem states that the eigenvalues of a Hermitian matrix A of order n are interlaced with those of any principal submatrix of order n − 1. Theorem 1 (Cauchy Interlace Theorem). Let A be a Hermitian matrix of order n, and let B be a principal submatrix of A of order n − 1 .I fλn ≤ λn−1 ≤ · ·· ≤λ2 ≤ λ1 lists the eigenvalues of A and µn ≤ µn−1 ≤ · ·· ≤µ3 ≤ µ2 the eigenvalues of B, then λn ≤ µn ≤ λn−1 ≤ µn−1 ≤ · ·· ≤λ2 ≤ µ2 ≤ λ1.