Consider the solutions of the matrix inverse problem, which are symmetric positive semide finite on a subspace. Necessary and su f ficent conditions for the solvability, as well as the general solution are obtained. The best approa'imate solution by the above solution set is given. Thus the open problem in [1] is solved.

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。

In this paper, we consider the following two problems: Problem I. Given X E Rmxn, A=diag (λ1,•••,λm)>0, find A E BSRnoxn such that ‖AX-XA‖=min, where ‖•‖ is Frobenius norm, BSRnoxn is the set of all n x n bisymmetric nonneg-ative definite matrices. Problem II. Given A* ∈ Rnxn, find ALS ∈ SE such that ‖A*-ALS‖=inf ‖A*-A‖, AESE where SE is the solution set of problem I. The existence of the solution for problem I, Ⅱ and the uniqueness of the solution for Problem Ⅱ are proved. The general form of SE is given and the expression of ALS is presented.

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.