Semidefiniteness without real symmetry Original Research Article

Let A be an n-by-n matrix with real entries. We show that a necessary and sufficient condition for A to have positive semidefinite or negative semidefinite symmetric part image is that rank[H(A)X]less-than-or-equals, slantrank[XTAX] for all image . Further, if A has positive semidefinite or negative semidefinite symmetric part, and A2 has positive semidefinite symmetric part, then rank[AX]=rank[XTAX] for all image . This result implies the usual row and column inclusion property for positive semidefinite matrices. Finally, we show that if A,A2,…,Ak(kgreater-or-equal, slanted2) all have positive semidefinite symmetric part, then rank[AX]=rank[XTAX]=cdots, three dots, centered=rank[XTAk−1X] for all image