In this paper, we first consider the inverse problem as follows: Given two matrices X and B, find a matrix A such that AX = B, where A is a symmetrizable nonnegative definite matrix. The sufficient and necessary conditions are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by S_E. Then the approximation problem for the inverse problem is discussed. That is: Given an arbitrary A^*, find a matrix which is nearest to A^* in the Frobenius norm. We show that the best approximation is unique and provide an expression for this nearest matrix.

symmetrizable nonnegative definite matrices, inverse problem, matrix nearness problem, matrix norm.