We study dual number symmetric matrices, dual complex Hermitian matrices and dual quaternion Hermitian matrices in a unified frame of dual Hermitian matrices. Suppose we have a ring, which can be the real field, the complex field, or the quaternion ring. Then an $n \times n$ dual Hermitian matrix has $n$ dual number eigenvalues. We define 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 apply 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. Applications to low rank approximation and generalized inverses of dual matrices, dual least squares problem and formation control are discussed. Numerical experiments are reported.

翻译：我们以对偶埃尔米特矩阵的统一框架研究对偶数对称矩阵、对偶复埃尔米特矩阵和对偶四元数埃尔米特矩阵。假设存在一个环，可以是实数域、复数域或四元数环，则一个$n \times n$对偶埃尔米特矩阵具有$n$个对偶数特征值。我们为对偶埃尔米特矩阵定义了补充矩阵，补充矩阵为原环中的埃尔米特矩阵。该对偶埃尔米特矩阵特征值的标准部分由原环中标准部分埃尔米特矩阵的特征值构成，而特征值的对偶部分则由这些补充矩阵的特征值构成。因此，通过应用原环中埃尔米特矩阵特征值计算的任何实用方法，我们即可获得对偶埃尔米特矩阵特征值计算的实用方法，并将该方法称为补充矩阵法。本文讨论了该方法在对偶矩阵低秩近似与广义逆、对偶最小二乘问题及编队控制中的应用，并报告了数值实验结果。