Constrained Solutions Of Some Matrix Equations

In this paper,we consider the following three types of matrix constraint problems:Find the constrained solution of the inverse problem of 2× 2 block linear matrix equations and the corresponding solution of the best approximation problem.Find the Re-nonnegative definite and Re-positive definite solutions of the matrix equation AXB=C on a linear manifold.Find the Re-nonnegative definite and Re-positive definite generalized inverses of a given complex matrix.For solving the constrained solution problem of inverse 2×2 block matrix equation,the existence conditions and the expressions of the general solutions for the matrix equation(?)with respect to unknown matrices A and B are given by using the generalized singular value decomposition of matrix.Furthermore,the expressions of the general solutions for the inverse problem of the matrix A under different constraints are discussed.And the unique solution of the optimal approximation problem with a given matrix is obtained by using matrix derivatives.By using matrix decompositions,we obtain the necessary and sufficient conditions for the solvability and the expression of the general solution of the following matrix inequality A+BX+(BX)~*+CYD+(CYD)~*≥H(>H).And by using the results obtained,we get the explicit expressions for the general Renonnegative definite and Re-positive solutions of the equation AXB=C on a linear manifold and the explicit expressions for the Re-nonnegative definite and Re-positive definite inverses of a given complex matrix.