中文摘要：

讨论如下广义特征值反问题；给定矩阵X，对角阵A和广义反射阵P，求自反阵A，B使得AX=BXA，给出了（A，B）的一般表达式．我们把上述问题解的全体记为SAB。然后，讨论了上述问题的最佳逼近问题：给定任意矩阵A^＊，B^＊，求矩阵（A^-,B^-）∈SAB，使得在F-范数意义下（A^-,B^-）为（A^＊,B^＊）的最佳逼近．证明了此问题有惟一解，并给出解的表达式，算法及数值例子．

英文摘要：

In this paper, we first consider the inverse generalized eigenvalue problem as follows： Given a matrix X,a diagonal matrix A and a generalized reflection matrix P, the reflexive matrix solutions （A,B） of AX=BXA are considered. The general representation of such a solution is presented. We denote the set of such solutions by SAB. Then the optimal approximation problem is discussed. That is： given arbitrary matrices A^＊ ,B^＊ , find matrix（A^-,B^-）∈ SAB which is nearest to （A^＊ ,B^＊ ） in the Frobenius norm. We show that the optimal approximation solution is unique and provide an expression for the nearest matrices. The algorithm and one numerical example for solving optimal approximation solution are included.