Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices

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. If λ n ≤ λ n−1 ≤ · · · ≤ λ 2 ≤ λ 1 lists the eigenvalues of A and µ n ≤ µ n−1 ≤ · · · ≤ µ 3 ≤ µ 2 the eigenvalues of B, then λ n ≤ µ n ≤ λ n−1 ≤ µ n−1 ≤ · · · ≤ λ 2 ≤ µ 2 ≤ λ 1. Proofs of this theorem have been based on Sylvester's law of inertia [3, p. 186] and the Courant-Fischer minimax theorem [1, p. 411], [2, p. 185]. In this note, we give a simple, elementary proof of the theorem by using the intermediate value theorem. A = a y * y B , where * signifies the conjugate transpose of a matrix. Let D = diag(µ 2 , µ 3 ,. .. , µ n). Then, since B is also Hermitian, there exists a unitary matrix U of order n − 1 such that U * BU = D. Let U * y = z = (z 2 , z 3 ,. .. , z n) T. We first prove the theorem for the special case where µ n < µ n−1 < · · · < µ 3 < µ 2