Inverse Problem For Several Kinds Of Special Matrix Equations And Their Approximations

The constrained matrix equation and its least-squares solution problems have been widely used in structural design, biology, electrical, structural dynamics, solid mechanics, automatic control, vibration theory, nonlinear programming, dynamic analysis and so on. The research works on constrained matrix equations have significant theoretical and practical value. In this paper, we study the following problems:ProblemⅠGiven A∈R^m×n,B∈R^n×p,C∈R^m×p. S (?)R^n×nis a special matrix. Find X∈S^n×n, such that AXB =C.ProblemⅡGiven A∈R^m×n,B∈R^n×p,C∈R^m×p. S ?R^n×nis a special matrix. Find X∈S^n×n, such that ||AXB- C||=min.ProblemⅢGiven X *∈R^n×n.Find X(?)∈SE, such that where SE is the solution set of ProblemⅠorⅡ. ProblemⅣGiven X∈R^n×p, B∈R^p×p, S (?) R^n×n.FindA∈S, such that X^T AX=B.ProblemⅤGiven A *∈R^n×n. Find A(?)∈SE, such that AAAAwhere SE is the solution set of ProblemⅣ.The main results of this paper are as follows:1. The first three problems for the matrix equation AXB =C,(1) When S is the set all central symmetry matrices, we have studied ProblemⅠand ProblemⅡby using the generalized singular value decomposition(GSVD) and the canonical correction decomposition(CCD). We given some arithmetic and numeric example. (2) When is the set all bisymmetric matrices, we have studied ProblemⅠ, ProblemⅡand ProblemⅢ, obtained the solvability conditions and the expressions of S solutions of this three problems. And given some arithmetic and numeric example for ProblemⅢ.(3) When S is the set all symmetric and sub-anti-symmetric matrices, we have studied ProblemⅠand ProblemⅢ, and obtained the solvability conditions and the expressions of solutions of this two problems.2. The last two problems for X^T AX= B, matrix equation X^T AX=B can be considered as a special case of AXB = C, we have discussed the symmetric and sub-anti-symmetric problem, central symmetry problem and the least-squares approximation solutions by using GSVD.