On Wielandt-Mirsky's conjecture for matrix polynomials

Công-Trình Lê

Published 2018-11-08Version 1

In matrix analysis, the Wielandt-Mirsky's conjecture states that $$ dist(\sigma(A), \sigma(B)) \leq \|A-B\|, $$ for any normal matrices $ A, B \in \mathbb C^{n\times n}$ and any operator norm $\|\cdot \|$ on $C^{n\times n}$. 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 \textit{false} in general. However it is true for the Frobenius distance and the Frobenius norm (the Hoffman-Wielandt's inequality). The main aim of this paper is to study the Hoffman-Wielandt's inequality and some weaker versions of the Wielandt-Mirsky's conjecture for matrix polynomials.