The Solutions Of The Matrix Equation A～TXA=B On The Linear Manifolds

This thesis is devoted to the solutions of matrix equation A^TXA = B on some linear manifolds. It consists of six chapters.In Chapter 1, the background and present conditions are introduced and summarized for the study of the solutions of matrix equation A^TXA — B.In Chapter 2, the D-symmetric solutions of matrix equation A^TXA = B on the linear manifold S = {X ∈ D^-2SR^n×n|XZ = Y, Y = Y{D²Z)+D²Z, Z^TD²Y = Y^TD²Z, Y,Z∈ R^n×k}, by applying the generalized singular value decomposition (GSVD) of a matrix pair, the necessary and sufficient conditions for the existence and the expressions for the D-symmetric solutions of matrix equation A^TXA — B on the linear manifold are established. In addition,the least-squares solutions of A^TXA = B with respect to D-symmetric matrix are derived.In Chapter 3, the generalized skew-symmetric solutions of the matrix equation A^TXA = B on the linear manifold S = {X e GKSR^n×n|||XY - Z}}= min, Y, Z∈ R^n×m}, using the canonical correlation decomposition of a matrix pair, the necessary and sufficient conditions for the existence and the expressions for the generalized skew-symmetric solutions of matrix equation A^TXA — B on the linear manifold are established. The expression of the least-squares solutions of the matrix equation A^TXA — B on the linear manifold is derived, too. In addition, by applying the quotient singular value decomposition of a matrix pair, we obtained a general expression of the least-squares solutions of the matrix equation A^TXA — B on the linear manifold.In Chapter 4, the centro-symmetric solutions of matrix equation A^TXA = B on the linear manifold S = {X ∈ CSR^n×n|||XY - Z|| = min, Y,Z ∈ R^n×m|}, using the quotient singular value decomposition(QSVD) of matrix pairs, the necessary and sufficient conditions for the existence and the expressions for the centro-symmetric solutions of matrix equation A^TXA = B on the linear manifold are established. In addition, in the solution set of corresponding equation, the expression of the optimalapproximation solution to given matrix is derived.In Chapter 5, the least-squares solutions bisymmetric matrix set of the matrix equation ATXA = B on two linear manifolds Si = {X € BSRnxn\\\XY - Z\\ = min} and S2 = {X € BSRnXn\XY = Z, YtTZi = ZjYu ZtffYi = Zi,i = 1,2, Y, Z e Rnxm}, by applying the singular value decomposition of matrix and the canonical correlation decomposition of matrix pairs, we obtain a general expression of the least-squares solutions of the matrix equation ATXA = B on two linear manifolds.In Chapter 6, the symmetric ortho-symmetric solution of the matrix equation AXAT = Bon the linear manifold S = {X 6 SPRnxn\XZ = Y, ZjYt = Y?Zh YtZ+Zi = Yt, i = 1,2, Y,Ze Rnxm}, by making use of the quotient singular value decomposition (QSVD) of a matrix pair, we established the necessary and sufficient conditions for the existence of and the expressions for the symmetric ortho-symmetric solution of minimum Frobenius norm of matrix equation AXAT = B...