线性流形上的广义反射矩阵反问题

Inverse Problems for Generalized Reflexive Matrices on a Linear Manifold

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AX_2=Z_2,Y_2~HA=W_2~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.

Let R ∈Cm×m and S ∈Cn×n be nontrivial unitary involutions, i.e., RH=R=R-1 ≠ ± Im and SH=S=S-1 ≠ ± In. A ∈Cm×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, Y2H A=W2H 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.