Abstract:

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.

Key words: Matrix equation, Part Hermitian solution, Optimal approximation