Eigenvalues of Dual Hermitian Matrices with Application in Formation Control

TL;DR

This paper introduces a supplement matrix method for efficiently computing eigenvalues of dual Hermitian matrices, with applications in multi-agent formation control and a connection to unit gain graph theory.

Contribution

It develops a unified framework for dual Hermitian matrices over various rings and links eigenvalues to supplement matrices, aiding formation control analysis.

Findings

Eigenvalues of dual Hermitian matrices can be computed via supplement matrices.

The method applies to real, complex, and quaternion rings.

Numerical experiments validate the proposed approach.

Abstract

We propose a supplement matrix method for computing eigenvalues of a dual Hermitian matrix, and discuss its application in multi-agent formation control. Suppose we have a ring, which can be the real field, the complex field, or the quaternion ring. We study dual number symmetric matrices, dual complex Hermitian matrices and dual quaternion Hermitian matrices in a unified frame of dual Hermitian matrices. An $n \times n$ dual Hermitian matrix has $n$ dual number eigenvalues. We define determinant, characteristic polynomial and supplement matrices for a dual Hermitian matrix. Supplement matrices are Hermitian matrices in the original ring. The standard parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of the standard part Hermitian matrix in the original ring, while the dual parts of the eigenvalues of that dual Hermitian matrix are the eigenvalues of those supplement matrices. Hence, by applying any practical method for computing eigenvalues of Hermitian matrices in the original ring, we have a practical method for computing eigenvalues of a dual Hermitian matrix. We call this method the supplement matrix method. In multi-agent formation control, a desired relative configuration scheme may be given. People need to know if this scheme is reasonable such that a feasible solution of configurations of these multi-agents exists. By exploring the eigenvalue problem of dual Hermitian matrices, and its link with the unit gain graph theory, we open a cross-disciplinary approach to solve the relative configuration problem. Numerical experiments are reported.