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.KeywordsNormal MatrixHermitian MatrixSpectral VariationUnitary MatriceHermitian MatriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.