In this paper, the inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their associated optimal approximation problems which are constrained by their partially prescribed eigenpairs are considered, respectively. First, the necessary and sufficient conditions of the solvability for the inverse eigenvalue problems of skew-Hermitian reflexive and anti-reflexive matrices are both derived, and the general solutions are also presented. Then the solutions of the corresponding optimal approximation problems in the Frobenius norm to a given matrix are also given, respectively. Furthermore, we give the algorithms to compute the optimal approximate skew-Hermitian reflexive and anti-reflexive solutions and present some illustrative numerical examples.

在本文中，分别考虑了倾斜-Hermitian自反矩阵和反自反矩阵的特征值反问题及其相关的最佳逼近问题，这些问题受其部分规定的特征对约束。首先，推导了偏Hermitian自反矩阵和反自反矩阵特征值反问题的可解性的充要条件，并给出了一般解。然后，还给出了Frobenius范数中给定矩阵的相应最佳逼近问题的解。此外，我们给出了计算最佳近似偏斜-Hermitian自反和反自反解的算法，并给出了一些说明性的数值示例。