矩阵方程AX=B的一类反问题及数值解法

A CLASS OF INVERSE PROBLEMS OF MATRIX EQUATION AX=B AND ITS NUMERICAL SOLUTION

§1.引言 用I_r表示r阶单位阵,R^(n×m)表示所有n×m实矩阵的集合.||·||_F表示Frobenius范数.若?0≠x∈R^n有x^TAx≥0(>0),则记为A≥0(>0);若A≥0(>0)且A=A^T,则称A为对称半正定(正定)阵.

This paper proves a theorem for judging if a matrix is a semi-positive definitematrix. With this theorem, the sufficient and necessary condition under which S isnonempty is obtained where S = {A∈ R^(n×n)} A = A^T, AX = B, X∈R^(n×m), B∈R^(n×m),x^TAx≥0, ?x∈R^n}. The general form of S is given. A numerical method forfinding the optimal approximation solution of the inverse problem of matrices isprovided.