Eigenvalues of Dual Hermitian Matrices with Application in Formation Control
Liqun Qi, Chunfeng Cui · SIAM Journal on Matrix Analysis and Applications · 2024
Abstract. We propose a supplement matrix method for computing eigenvalues of a dual Hermitian matrix and discuss its application in multiagent 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 [Formula: see text] dual Hermitian matrix has [Formula: see text] 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 of the dual 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. Note that the standard part of the dual Hermitian matrix is still a Hermitian matrix in the original ring. 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 multiagent 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 multiagents 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 to show the efficiency of our proposed methods.