Factorization of Matrices of Quaternions

TL;DR

This paper reviews and derives various matrix factorizations for quaternion matrices, using complex matrix representations with specific symmetries, and discusses related spectral theorems without focusing on algorithms.

Contribution

It provides a comprehensive derivation of classical matrix factorizations for quaternion matrices using complex symmetric representations, and establishes a Schur and spectral theorem for commuting matrices.

Findings

Derived Jordan canonical form for quaternion matrices

Established polar decomposition and SVD in the quaternion context

Proved Schur and spectral theorems for commuting quaternion matrices

Abstract

We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature. Rather than work directly with matrices of quaternions, we work with complex matrices with a specific symmetry based on the dual operation. We discuss related results regarding complex matrices that are self-dual or symmetric, but perhaps not Hermitian.