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.

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.

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.

A class of inverse eigenpair problem is proposed for real symmetric positive definite Jacobi mtrices. Necessary and sufficient conditions for the existence of a unique solution of this problem, as well as the analytic formula of this solution are derived. A numerical algorithm for computing the solution is also presented.