线性流形上的广义反射矩阵反问题

设 R∈C^m×m 及 S∈C^n×n 是非平凡Hermitian酉矩阵, 即 R^H=R=R^-1≠±I_m ,S^H=S=S^-1≠±I_n.若矩阵 A∈C^m×n 满足 RAS=A, 则称矩阵 A 为广义反射矩阵.该文考虑线性流形上的广义反射矩阵反问题及相应的最佳逼近问题.给出了反问题解的一般表示, 得到了线性流形上矩阵方程AX₂=Z₂, Y₂^H A=W₂^H 具有广义反射矩阵解的充分必要条件, 导出了最佳逼近问题唯一解的显式表示.

Inverse Problems for Generalized Reflexive Matrices on a Linear Manifold

Let R ∈C^m×m and S ∈C^n×n be nontrivial unitary involutions, i.e., R^H=R=R^-1 ≠ ± I_m and S^H=S=S^-1 ≠ ± I_n. A ∈C^m×n is said to be a generalized reflexive matrixif RAS=A.  This paper is concerned with the inverse problem for generalized reflexive matrices on a linear manifold and the optimal approximation to a given matrix. The general expression of the solutions of the problem is presented. Sufficient and necessary conditions for equations AX₂=Z₂, Y₂^H A=W₂^H having a common generalized reflexive matrix solution on the linear manifold are derived. The expression of the solution for relevant optimal approximation problem is given.