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

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 $D$.