The Power Method for Non-Hermitian Dual Quaternion Matrices
Hao Yang, Liqun Qi, Chunfeng Cui
TL;DR
This work develops a power method for computing the dominant eigenvalue of non-Hermitian dual quaternion matrices, overcoming challenges posed by potential nonexistence or multiplicity of eigenvalues via a Jordan-like decomposition. It establishes a sufficient (and necessary) condition for eigenvalue existence and proves linear convergence to the strict dominant eigenpair under this condition, with explicit dual-norm rate bounds. The authors extend the framework to non-Hermitian dual complex matrices and introduce a dual complex adjoint-based variant (DCAM-PM), providing comparable convergence behavior under stricter assumptions. Numerical experiments on formation-control Laplacians demonstrate the method’s efficiency and illustrate how graph structure affects convergence, with code available for broader use.
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.