Inverse Problems For Several Kind Of Constrained Matrix

This thesis considers several inverse problems of constrained matrix and their best approximation. It consists of six chapters.In chapter 1,the background and present conditions are introduced and summarized for the study of inverse problems of constrained matrix and their best approximation.In chapter 2, we discuss the solvablity on the Hermite generalized Hamilton solutions of matrix equation ||AX-Z||²+||Y^TA-W^T||² =min by applying the singulau value decomposition of a matrix, obtain the necessary and sufficient conditions for the existence and the expressions for the Hermite generalized Hamilton solutions of matrix equation ||AX -Z||² +||Y^TA-W^T||² =min.In addition , in the solution set of corresponding equation,the expression of the optimal approximation solution to given matrix is derived.In chapter 3, we study the Hermite generalized Hamilton solutions of the matrix equation||AX₂-B₂|| = min on the linear manifold S = {A∈HHC^n×n|AX₁ =B₁, B₁₂X₁₁⁺ X₁₁= B₁₂, B₁₁X₁₂⁺X₁₂= B₁₁ , X₁₁^HB₁₁ =B₁₂^HX₁₂}, give the expressions for the Hermite generalized Hamilton solutions of the matrix equation||AX₂-B₂|| = min, In addition , in the solution set of corresponding equation, the expression of the optimal approximation solution to given matrix is derived.In chapter 4, we study the anti-bisymmetric solutions of matrix equation AXA^T=B on the linear manifold S = {X∈ABSR^n×n |||XY-Z|| = min}, using the quotient singula value decomposition(QSVD)of matrix pairs , obtain the necessary and sufficient conditions for the existence and the express-ions for the anti-bisymmetric solutions of matrix equation AXA^T=B . In addition , in the solution set of corresponding equation, the expression of the optimal approximation solution to given matrix is derived .In chapter 5, we disuss the necessary and sufficient conditions for the existence and the expressions for the positive definite (semidefinite) symmetrizable solutions of matrix equation (?) = min, and derive the optimal approximation semidefinite symmetrizable solutions to given matrices.In Chapter 6, we consider the D-anti-symmetric solutions of the matrix equation f(A) = ||AY-Z|| =min on the linear manifold S={A∈D^-2ASR^n×n||AX-B||=min, X,B∈R^n×m}, by applying the singulau value decomposition of a matrix, obtain the necessary and sufficient conditions for the existence and the expressions of the general Solution for the inverse problem of these matrices. For any n-by-n real matrix, it's optimal approximation was obtained.