Positive Semidefinite Partitioned Matrices and a LinearMatrix Eqnation and its Inverse Problem

Abstract:

      A real (may not symmetric) n×n matrix M is said to be positive semidefiniteif, for any real nonzero n diniensional row vector x, xMx^t≥0. In this paper, we givea necessary and sufficient condition for determining whether a partitioned n ×n matrixis positive semidefinite. Moreover, we consider the solutions of linear matrix equationAX = B with variant conditions on the solutions. The necessary and sufficient conditionsfor the consistency of this equation are derived using the canonical form of a matrix under the equivalence. The iverse problem of this equation with variant couditions on the solutions is also included.