Spectral Variation of Normal Matrices

Abstract

Let A be an n x n Hermitian matrix, and let λ↓₁ (A) ≥ λ↓₂ (A) ≥ … ≥ λ↓_n (A) be the eigenvalues of A arranged in decreasing order. In Chapter III we saw that λ↓_j (A), 1 ≤ j ≤ n , are continuous functions on the space of Hermitian matrices. This is a very special consequence of Weyl’s Perturbation Theorem: if A, B are two Hermitian matrices, then.

In turn, this inequality is a special case of the inequality (IV.62), which says that if Eig↓ (A) denotes the diagonal matrix with entries λ↓_j (A) down its diagonal, then we have for all Hermitian matrices A, B and for all unitarily invariant norms.