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.

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, four inverse eigenproblems with given three elgenvalues and cor-responding eigenvectors are considered, some necessary and sufficient conditions under which there exists a unique solution for these problems are given. Further-more some numerical algorithms and some numerical experiments are given.

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.

Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered, a new necessary and sufficient condition for solvablity is given, and the expression of solution is derived in the some special cases. Based on the ex-pression, the least spuares solution of an inverse eigenvalue problem for positive semidefinite symmetric matrices is also given.