Stability of quaternion matrix polynomials

TL;DR

This work develops a quaternionic analogue of stability theory for right quaternion matrix polynomials by linking stability to the complex adjoint polynomial $P_χ$ and by establishing ball-centered stability reductions. It provides eigenvalue localization between two concentric balls, and a quaternion Eneström–Kakeya-type bound, extending key complex results to the noncommutative setting. The authors identify polynomial classes where stability and hyperstability coincide and extend the framework to multivariate quaternion polynomials, deriving practical sufficient conditions for hyperstability from multivariate stability. These contributions enable concrete bounds on right eigenvalues and broaden the applicability of stability concepts to quaternionic polynomial problems.

Abstract

A right quaternion matrix polynomial is an expression of the form $P(λ)= \displaystyle \sum_{i=0}^{m}A_i λ^i$, 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 $P(λ)$ 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 $P(λ)$ 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 $P(λ)$ 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 $P(λ)$ lie between two concentric balls of specific radii in the set of quaternions centered at the origin. A generalization of the Enestr{ö}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.

Theorems & Definitions (60)

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