The Power Method for Non-Hermitian Dual Quaternion Matrices

TL;DR

This paper introduces a power method tailored for non-Hermitian dual quaternion matrices, providing theoretical guarantees for convergence and extending the approach to related matrix types, with numerical validation.

Contribution

It develops a novel power method for non-Hermitian dual quaternion matrices, including a new Jordan-like decomposition and convergence analysis, addressing complexities unique to dual quaternion eigenproblems.

Findings

Convergence of the power method under specific conditions

Existence of eigenvalues linked to the standard part matrix

Numerical experiments demonstrating efficiency

Abstract

This paper proposes a power method for computing the dominant eigenvalues of a non-Hermitian dual quaternion matrix (DQM). Although the algorithmic framework parallels the Hermitian case, the theoretical analysis is substantially more complex since a non-Hermitian dual matrix may possess no eigenvalues or infinitely many eigenvalues. Besides, its eigenvalues are not necessarily dual numbers, leading to non-commutative behavior that further complicates the analysis. We first present a sufficient condition that ensures the existence of an eigenvalue whose standard part corresponds to the largest magnitude eigenvalue of the standard part matrix. Under a stronger condition, we then establish that the sequence generated by the power method converges linearly to the strict dominant eigenvalue and its associated eigenvectors. We also verify that this condition is necessary. The key to our analysis is a new Jordan-like decomposition, which addresses a gap arising from the lack of a conventional Jordan decomposition for non-Hermitian dual matrices. Our framework readily extends to non-Hermitian dual complex and dual number matrices. We also develop an adjoint method that reformulates the eigenvalue problem into an equivalent form for dual complex matrices. Numerical experiments on non-Hermitian DQMs are presented to demonstrate the efficiency of the power method.