一类线性流形上矩阵方程AX=B的反问题

The Inverse Problems of Matrices Equation AX=B on A Class of Linear Manifold

设Ω={A∈ASRn×n|Ax=C,x∈■T(S),SS+C=0,T2TC2=-C2TT2,C2T2+T2=C2},考虑问题Ⅰ:给定X,B∈Rn×m,求使得‖AX-B‖=min,A∈Ω的解集合SE;问题Ⅱ:给定A*∈Rn×n,求∈SE,使得‖A*-‖=minA∈SE‖A*-A‖。本文给出了问题Ⅰ、Ⅱ的解的通式。

Let Ω= {A∈ASR^n×n｜Ax=C, A x∈RT（S） ,SS^＋ C=0, T2^TC2 = -C2TT2 ,C2 T2^＋ T2 =C2 }, This paper considers the problem Ⅰ： Given X,B ∈ R^n×m, find A ∈ Ω, such that ｜｜AX-B｜｜= min; and considers the problem ｜｜ ： Given A^＊ ∈ R^n×n, find A ∈ SE, where SE = {A ｜ A ∈ Ω, ｜｜AX-B｜｜ = min}, such that ｜｜A^＊ -A ｜｜ = min A ∈ SE｜｜A^＊ -A ｜｜. The expressions for general solutions of the Problem Ⅰ、Ⅱ are given.