von Neumann type trace inequality for dual quaternion matrices

TL;DR

This paper extends classical matrix inequalities to dual quaternion matrices, providing new spectral norm bounds and perturbation inequalities that are useful for matrix theory and applications in multi-agent systems.

Contribution

It introduces a von Neumann type trace inequality and Hoffman-Wielandt inequality specifically for dual quaternion matrices, expanding the theoretical framework.

Findings

Established a spectral norm concept for dual quaternion matrices.

Derived a von Neumann type trace inequality for dual quaternion matrices.

Presented a Hoffman-Wielandt type inequality for singular values.

Abstract

Dual quaternion matrices have important applications in multi-agent formation control. In this paper, we first address the concept of spectral norm of dual quaternion matrices. Then, we introduce a von Neumann type trace inequality and a Hoffman-Wielandt type inequality for general dual quaternion matrices, where the latter characterizes a simultaneous perturbation bound on all singular values of a dual quaternion matrix. In particular, we also present two variants of the above two inequalities expressed by eigenvalues of dual quaternion Hermitian matrices. Our results are helpful for the further study of dual quaternion matrix theory, algorithmic design, and applications.