On Wielandt-Mirsky's conjecture for matrix polynomials

Abstract

In matrix analysis, the \textit{Wielandt-Mirsky conjecture} states that $$dist(\sigma (A),\sigma (B))\le \Vert A-B\Vert ,$$ for any normal matrices $A,B\in {\mathbb{C}}^{n\times n}$ and any operator norm $\Vert \cdot \Vert$ 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 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.

11 pages, a version of a Wielandt's inequality for matrix polynomials added, to be published in BKMS