Stability of quaternion matrix polynomials

TL;DR

This paper extends stability analysis of complex matrix polynomials to quaternion matrices, establishing eigenvalue locations and generalizing classical theorems for quaternion polynomial stability.

Contribution

It introduces methods to determine quaternion polynomial stability, relates quaternion stability to complex adjoint matrices, and generalizes the Eneström-Kakeya theorem.

Findings

Eigenvalues lie within specific concentric quaternion balls.

Stability with respect to a quaternion ball relates to complex stability.

Quaternion polynomial stability and hyperstability are shown to be equivalent in certain classes.

Abstract

A right quaternion matrix polynomial is an expression of the form $P(\lambda)= \displaystyle \sum_{i=0}^{m}A_i \lambda^i$, where $A_i$'s are $n \times n$ quaternion matrices with $A_m \neq 0$. The aim of this manuscript is to determine the location of right eigenvalues of $P(\lambda)$ relative to certain subsets of the set of quaternions. In particular, we extend the notion of (hyper)stability of complex matrix polynomials to quaternion matrix polynomials and obtain location of right eigenvalues of $P(\lambda)$ using the following methods: $(1)$ we give a relation between (hyper)stability of a quaternion matrix polynomial and its complex adjoint matrix polynomial, $(2)$ we prove that $P(\lambda)$ is stable with respect to an open (closed) ball in the set of quaternions, centered at a complex number if and only if it is stable with respect to its intersection with the set of complex numbers and $(3)$ as a consequence of $(1)$ and $(2)$, we prove that right eigenvalues of $P(\lambda)$ lie between two concentric balls of specific radii in the set of quaternions centered at the origin. A generalization of the Enestr{\"o}m-Kakeya theorem to quaternion matrix polynomials is obtained as an application. We identify classes of quaternion matrix polynomials for which stability and hyperstability are equivalent. We finally deduce hyperstability of certain univariate quaternion matrix polynomials via stability of certain multivariate quaternion matrix polynomials.