Quasi-determinant and right eigenvalues of dual quaternion matrices

TL;DR

This paper explores properties of dual quaternion matrices, introducing quasi-determinants and linking eigenvalues to characteristic polynomials, with implications for applications in brain science and multi-agent systems.

Contribution

It introduces the concept of quasi-determinants for dual quaternion matrices and establishes their relation to eigenvalues and characteristic polynomials.

Findings

Eigenvalues of dual quaternion matrices are roots of their quasi-characteristic polynomials.

Quasi-determinant of a dual quaternion Hermitian matrix equals the product of squared eigenvalue magnitudes.

Properties like Sturm theorem and Bloomfield-Watson inequality are extended to dual complex matrices.

Abstract

Dual quaternion/complex matrices have important applications in brain science and multi-agent formation control. In this paper, we first study some basic properties of determinants of dual complex matrices, including Sturm theorem and Bloomfield-Watson inequality for dual complex matrices. Then, we show that every eigenvalue of a dual complex matrix must be the root of the characteristic polynomial of this matrix. With the help of the determinants of dual complex matrices, we introduce the concept of quasi-determinants of dual quaternion matrices, and show that every right eigenvalue of a dual quaternion matrix must be the root of the quasi-characteristic polynomial of this matrix, as well as the quasi-determinant of a dual quaternion Hermitian matrix is equivalent to the product of the square of the magnitudes of all eigenvalues. Our results are helpful for the further study of dual quaternion matrix theory, and their applications.