In this paper, a class of inverse problems of matrix equation AX=B is studied on the linear manifold S={A ∈ SR×n‖AZ-Y‖=min}, the necessary and sufficient conditions for the olvability of the inverse problem and the expression of the general solution are given; at the same time, the best approximation problem is considered, the expression of the best approximate solution and the numerical method are also given. This paper extends the results in [1, 2].

Lets={A ∈ SRn×n ｜‖ AZ-Y‖=min} where Z,Y ∈ Rn×k, SRnxn={A ∈ Rn×n] AT = A}, ‖.‖ is the Frobenius norm. We consider the following problems: Problem I. Given X, B ∈ Rn×m, find A ∈ S ∩ Pn such that AX=B, where Pn={A ∈ SRn×n ｜Ax ∈ Rn, xTAx ≥0}. oblem II. Given A ∈ Rn×n, find A ∈ SE, such that ‖A-A‖=inf ‖A-A‖, VAESE where SE is the solution set of Problem I. The sufficient and necessary condition under which SE is nonempty is obtained. The general form of SE is given. Then expression of the solution A of problem II is presented and the numerical method is described.

In this paper we give some sufficient conditions and some necessary conditions under which the matrix equation X + A*X-nA=I has a positive deft-nite solution. An iterative method which converges to a positive definite solution of this equation is constructed. And an error estimate formula on this iterative method is also derived.

对于任意给定的矩阵A ∈Rkxn，B ∈Rkxn 和C ∈Rkxk，利用奇异值分解和广义奇异分解，我们给出了矩阵方程AXAT+BYBT=C的对称与反对称最小范数最小二乘解的表达式。

By applying the canonical correlation decomposition (CCD) of matrix pairs, we obtain a general expression of the least-squares solutions of the matrix equa-tion ATXA = D under the restriction that the solution matrix X∈ Rnxn is bisymmetric, where A ∈ Rnxm and D ∈ Rmxm are given matrices.