Eigenvalues and Singular Value Decomposition of Dual Complex Matrices

TL;DR

This paper extends eigenvalue and singular value decomposition concepts to dual complex matrices, establishing properties of Hermitian matrices and their diagonalization, with extensions to dual quaternion matrices.

Contribution

It introduces right eigenvalues and subeigenvalues for dual complex matrices and develops their spectral properties, including diagonalization criteria and SVD extension.

Findings

Hermitian dual complex matrices have real eigenvalues and subeigenvalues.

Diagonalization occurs if and only if no right subeigenvalues exist.

Results extend to dual quaternion matrices.

Abstract

We introduce right eigenvalues and subeigenvalues for square dual complex matrices. An $n \times n$ dual complex Hermitian matrix has exactly $n$ right eigenvalues and subeigenvalues, which are all real. The Hermitian matrix is positive semi-definite or definite if and only if all of its right eigenvalues and subeigenvalues are nonnegative or positive, respectively. A Hermitian matrix can be diagonalized if and only if it has no right subeigenvalues. Then we present the singular value decomposition for general dual complex matrices. The results are further extended to dual quaternion matrices.