$m$th roots of $H$-selfadjoint matrices over the quaternions

TL;DR

This paper establishes necessary and sufficient conditions for the existence of $H$-selfadjoint $m$th roots of quaternion matrices, providing a constructive method when such roots exist, using complex matrix representations.

Contribution

It introduces a complete characterization and construction method for $H$-selfadjoint $m$th roots of quaternion matrices, extending previous work to quaternionic settings.

Findings

Necessary and sufficient conditions for existence

Explicit construction method provided

Application of complex matrix representation

Abstract

The complex matrix representation for a quaternion matrix is used in this paper to find necessary and sufficient conditions for the existence of an $H$-selfadjoint $m$th root of a given $H$-selfadjoint quaternion matrix. In the process, when such an $H$-selfadjoint $m$th root exists, its construction is also given.