  In matrix analysis, the \textit{Wielandt-Mirsky 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 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.
