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 Further, if A has positive semidefinite or negative semidefinite symmetric part, and A 2 has positive semidefinite symmetric part, then rank[AX] = rank[X T AX] for all X ∈ M n (R). This result implies the usual row and column inclusion property for positive semidefinite matrices. Finally, we show that if A, A 2 , . . . , A k

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... 2) all have positive semidefinite symmetric part, then rank[AX] = rank[X T AX] = · · · = rank[X T A k−1 X] for all X ∈ M n (R).