Semidefiniteness without Hermiticity

Let A ∈ Mn(C ). We give a rank characterization of the semidefiniteness of Hermitian A in two ways. We show that A is semidefinite if and only if rank[X∗AX] = rank[AX], for all X ∈ Mn(C ), and we show that A is semidefinite if and only if rank[X∗AX] = rank[AXX∗], for all X ∈ Mn(C ). We show that if A has semidefinite Hermitian part and A has positive semidefinite Hermitian part then A satisfies row and column inclusion. Let B ∈ Mn(C ), and k an integer with k ≥ 2. If BBA,BBA, . . . , BBA each have positive semidefinite Hermitian part, we show that rank[BAX] = rank[X∗B∗BAX] = · · · = rank[XBBAX], for all X ∈ Mn(C ). These results generalize or strengthen facts about real matrices known earlier.