The Hoffman-Wielandt inequality for quaternion matrices and quaternion matrix polynomials

TL;DR

This paper extends the Hoffman-Wielandt inequality to quaternion matrices and polynomials, analyzing eigenvalue bounds and diagonalizability of associated block matrices for quadratic and linear cases.

Contribution

It introduces a generalized Hoffman-Wielandt inequality for quaternion matrices and polynomials, and investigates eigenvalue bounds for quaternion matrix polynomials with unitary coefficients.

Findings

Generalized Hoffman-Wielandt inequality for quaternion matrices and polynomials

Diagonalizability of block companion matrices of quaternion polynomials

Eigenvalue bounds for quaternion matrix polynomials with unitary coefficients

Abstract

The purpose of this paper is to derive the Hoffman-Wielandt inequality and its generalization for quaternion matrices. Diagonalizability of the block companion matrix of certain quadratic (linear) quaternion matrix polynomials is brought out. As a consequence, we prove that if $Q(\lambda)$ is another quadratic (linear) quaternion matrix polynomial, then under certain conditions on the coefficients, a generalization of the Hoffman-Wielandt inequality for their corresponding block companion matrices holds. We also prove that if $P(\lambda)$ is a quaternion matrix polynomial with unitary coefficients, then any right eigenvalue $\lambda_0$ of $P(\lambda)$ lies in the annular region $\frac{1}{2} < |\lambda_0| < 2$.