Quaternion matrix decomposition and its theoretical implications

TL;DR

This paper introduces a new quaternion matrix decomposition technique that improves upon previous methods and enables enhanced results in quaternion domain optimization problems.

Contribution

The paper presents a novel rank-one decomposition for quaternion Hermitian matrices with stronger properties than existing methods.

Findings

Improved results in joint numerical range analysis

Enhanced $ ext{S}$-Procedure applications in quaternion domain

Advancements in quaternion QCQP solutions

Abstract

This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in (sturm2003cones,huang2007complex,ai2011new). The enhanced property can be used to drive some improved results in joint numerical range, $\mathcal{S}$-Procedure and quadratically constrained quadratic programming (QCQP) in the quaternion domain, demonstrating the capability of our new decomposition technique.