Computing Left Eigenvalues of Quaternion Matrices

TL;DR

This paper introduces a Newton-based method for efficiently computing left eigenvalues of quaternion matrices, applicable to matrices of any size, with demonstrated accuracy and reproducibility through extensive testing.

Contribution

It presents a practical, generalizable algorithm leveraging standard linear algebra kernels for quaternion eigenvalue problems, including detection of complex spectral phenomena.

Findings

Effective for matrices up to 64x64 in size

Detects multiple spherical components and non-generic phenomena

Provides a reproducible MATLAB implementation

Abstract

We present a practical Newton-based method for computing left eigenvalues of quaternion matrices. It uses only standard real/complex linear-algebra kernels via embeddings and applies to matrices of any size. Extensive tests on literature examples and benchmark ensembles, together with a compact MATLAB reference implementation, demonstrate reproducible, certificate-based computations up to size 64x64, including the detection of multiple spherical components and non-generic phenomena such as more than n isolated left eigenvalues and left-spectrum deficiency.