The Inverse Problem Of Matrix And Its Best Approximation

The inverse problem of matrices have been widely used in control theory, vibration theory and civil struction engineering, economy field etc. We mainly discuss the following problems.Problem I : Given X, B e R ,S R .Find A e S such that: AX = B Problem II: Given X,B R ,S R .Find A e S such that: ||AX - B|| = minProblemIII: Given A R . Find A SE, such that:||A-A||F =min||A-A||FWhere SE is the solution set of Problem I or II .The main results of this paper are as follows:1. when S is all anti-symmetric matrices set, we have proved the existence of solution for Problem I .2. when S is all anti-bisymmetric matrices set, we have discussed the construction of this kind of matrices. And we have success proved the existence of solution for Problem I and Problem II and the uniqueness of solution for Problemlll. We have also obtained the general expression of solutions and we given the numerical example.3. when S is all symmetries and skew symmetrices matrices set, we have discussed the construction of this kind of matrices. And we have proved the existence of solution for Problem I and the unique of solution for Problem III. And on the linear manifold, we have discussed the existence of solution for Problem I and the unique of solution for Problemlll, and we given the numerical example.4. when S is all bisymmetric matrices, on the linear manifold, we have discussed the existence of solution for Problem I and the unique of solution for Problemlll, and we given the numerical example.This paper is supported by the Beijing special item outlay for elitist.