On Wielandt-Mirsky’s Conjecture for Matrix Polynomials

In matrix analysis, the Wielandt-Mirsky conjecture states that

dist(sigma(A), sigma(B)) <= parallel to A - B parallel to

for any normal matrices A, B is an element of C-nxn and any operator norm parallel to . parallel to on C-nxn. Here dist(sigma(A),sigma(B)) denotes the optimal matching distance between the spectra of the matrices A and B. It was proved by A. J. Holbrook (1992) that this conjecture is false in general. However it is true for the Frobenius distance and the Frobenius norm (the Hoffman-Wielandt inequality). The main aim of this paper is to study the Hoffman-Wielandt inequality and some weaker versions of the Wielandt-Mirsky conjecture for matrix polynomials.