The Constrained Solutions of two Matrix Equations

摘要

We study the symmetric positive semidefinite solution of the matrix equation AX_1A~T+ BX_2A~T=C, where A is a given real m×n matrix, B is a given real m×p matrix, and C is a given Real m ×m matrix, with m, n, p positive integers; and the bisymmetric positive semidefinite solution Of the matrix equation D~TXD=C, where D is a given real n×m matrix, C is a given real m×m Matrix, with m, n positive integers. By making use of the generalize singular value decomposition, we Derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the Existence of these solutions.