一类矩阵方程的广义Hermite问题

该文主要解决了如下两个问题 问题I 已知矩阵 M∈ C^n×e, A∈C^n×m, B∈ C^m×m, 求 X∈ HC_M,n使得 A^HXA=B, 其中 HC_M,n={ X∈ C^n×n}|α^H(X-X^H)=0, for all α∈ C(M) }. 问题II 任意给定矩阵 X^* ∈C^n×n, 求 $\hat{X}\in H_E$ 使得 ||\hat{X}-X^*||=\min\limits_{X∈ H_E}||X-X^*||, 这里 H_E 为问题I的解集. 利用广义奇异值分解定理,得到了问题I的可解条件及其通解表达式, 获得了问题II的解,并进行了相应的数值计算.

The Part Hermitian Solutions of Matrix Equation AHXA=B on Subspace

In this paper, the following two problems are considered: Problem I Given M∈ C^n×e, A∈C^n×m, B∈ C^m×m, find X∈ HC_M,n such that A^HXA=B, where HC_M,n={ X∈ C^n×n}|α^H(X-X^H)=0, for all α∈ C(M) }. Problem II Given X^* ∈C^n×n, find $\hat{X}\in H_E$ such that ||\hat{X}-X^*||=\min\limits_{X∈ H_E}||X-X^*||, where H_E is the solution set of Problem I. The necessary and sufficient condition for the solvability and the general form of the solutions Problem I are given. For Problem II, the expression for the solution, a numerical algorithm and a numerical example are illustrated.