Semidefiniteness without real symmetry

Let A be an n-by-n matrix with real entries. We show that a necessary and s ufficient condition for A to have positive semidefinite or negative semidef inite symmetric part H(A) = 1/2 (A + A(T)) is that rank[H(A)X] less than or equal to rank[X(T)AX] for all X is an element of M-n(R). Further, if A has positive semidefinite or negative semidefinite symmetric part, and A(2) ha s positive semidefinite symmetric part, then rank[AX] = rank[X(T)AX] for al l X is an element of M-n(R). This result implies the usual row and column i nclusion property for positive semidefinite matrices. Finally, we show that if A, A(2),..., A(k)(k greater than or equal to 2) all have positive semid efinite symmetric part, then rank[AX] = rank[X(T)AX] = (...) = rank[X(T)A(k -1)X] for all X is an element of M-n(R). (C) 2000 Elsevier Science Inc. All rights reserved.