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.

翻译：我们提出了一种补充矩阵方法用于计算对偶厄米矩阵的特征值，并探讨了该方法在多智能体编队控制中的应用。设存在一个环，可以是实数域、复数域或四元数环。我们在对偶厄米矩阵的统一框架下研究了对偶数对称矩阵、对偶复厄米矩阵和对偶四元数厄米矩阵。一个$n \times n$对偶厄米矩阵具有$n$个对偶数特征值。我们定义了对偶厄米矩阵的行列式、特征多项式和补充矩阵。补充矩阵是原始环中的厄米矩阵。该对偶厄米矩阵特征值的标准部分为原始环中标准部分厄米矩阵的特征值，而其特征值的对偶部分则是这些补充矩阵的特征值。因此，通过应用任意实用方法计算原始环中厄米矩阵的特征值，即可获得计算对偶厄米矩阵特征值的实用方法。我们将该方法称为补充矩阵法。在多智能体编队控制中，可能给定一个期望的相对构型方案。人们需要判断该方案是否合理，从而确保存在可行的多智能体构型解。通过探索对偶厄米矩阵的特征值问题及其与单位增益图理论的联系，我们开辟了一条解决相对构型问题的跨学科途径。文中还报告了数值实验。