Least-Squates Solutions Of Inverse Problems For Two Class Of Matrices And Their Optimal Approximation

The constrained matrix equations have been widely used in control theory, vibration theory, civil structure engineering and nonlinear program etc. This thesis studies systematically several types of constrained matrix equations. We are concerned with the following 4 problems:ProblemⅠ, Given X∈Rn×m,Λ=diag (λ1 ,,λm )∈Rm×m and S ? Rn×n. Find A∈S,such that AX = XΛ.ProblemⅡ,Given X , B∈Rn×m and S Rn×n. Find A∈S,such that AX=B.ProblemⅢ,Given X , B∈Rn×m and S Rn×n. Find A∈S,such that AX - B=min.ProblemⅣ,Given A *∈Rn×n. Find A?∈SE,such that A- A* minAA*? =A∈S E?. Here S E is the solution set of problemⅠ,ⅡorⅢ,and ||·|| is the Frobenius norm. Matrix set S can be any one of W -1 SRn×n or W -1 ASRn×n.The main results are obtained in this thesis are as follows:1. When S is the set of all W-para-symmetric matrices, we first discus the structure of S and analyse its properties. Then we prove the existence of the solution of ProblemsⅠ,ⅢandⅣ. We also propose a sufficient condition for ProblemⅣto have a unique solution. In addition, we derive a general expression for the solution of ProblemsⅠ,ⅢandⅣ.2. For the case when S is the set of all W-para-symmetric matrices, we consider the least squares solution of ProblemsⅢandⅣ, and their optimal approximation. We show the existence and uniqueness of the optimal approximation solution. We also derive expressions for optimal approximation solution and the least squares solution.3. When S is the set of all W-para-anti -symmetric matrices, we first discus the structure of S and analyze its properties. Then we prove the existence of the solution of ProblemsⅠ,ⅢandⅣ. We also propose a sufficient condition for ProblemⅣto have a unique solution. In addition, we derive a general expression for the solution of ProblemsⅠ,ⅢandⅣ.4. For the case when S is the set of all W-para-anti -symmetric matrices, we consider the least squares solution of ProblemsⅢandⅣ, and their optimal approximation. We show the existence and uniqueness of the optimal approximation solution. We also derive expressions for optimal approximation solution and the least squares solution.