Hoffman-Wielandt type inequality for block companion matrices of certain matrix polynomials

TL;DR

This paper establishes a Hoffman-Wielandt type inequality for block companion matrices of quadratic matrix polynomials with unitary or doubly stochastic coefficients, under certain diagonalizability conditions.

Contribution

It proves diagonalizability of block companion matrices for specific matrix polynomials and derives a Hoffman-Wielandt inequality for their eigenvalues.

Findings

Block companion matrices are diagonalizable under certain conditions.

A Hoffman-Wielandt type inequality is established for these matrices.

Results apply to quadratic matrix polynomials with special coefficient structures.

Abstract

Matrix polynomials with unitary/doubly stochastic coefficients form the subject matter of this manuscript. We prove that if $P(\lambda)$ is a quadratic matrix polynomial whose coefficients are either unitary matrices or doubly stochastic matrices, then under certain conditions on these coefficients, the corresponding block companion matrix $C$ is diagonalizable. Consequently, if $Q(\lambda)$ is another quadratic matrix polynomial with corresponding block companion matrix $D$, then a Hoffman-Wielandt type inequality holds for the block companion matrices $C$ and $D$.