Supplement Matrix and a Practical Method for Computing Eigenvalues of a Dual Hermitian Matrix

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.