Rocky Mountain Journal of Mathematics
Let A is an element of M(n)(C). We provode a rank characterization of the semidefiniteness of Hermitian A in two ways. We show that A is semidefinite if and only of rank [X* AX] = rank [AX], for all X is an element of M(n)(C), and that A is ssemidefinite if and only if rank [AXX*], for all X is an element of Mn(C). We show that, if A has semidefinite Hermitian part and A(2) has positive semidefinite Hermitian part, then A satisfies row and column inclusion. Let B is an element of M(n)(C), and let kappa be an integer with k >= 2. If B*BA, B* BA(2), ..., N* BA(k) each has positive semidefinite Hermitian part; we show that rank [NAX} = rank [X*B*BAX] = ... = rank [X*B*BA(kappa-1)X], for all X is an element of M(n)(C) These results generalize or strengthen facts about real matrices known earlier.
Johnson, Charles R.; Reams, Robert. Semidefiniteness without Hermiticity. Rocky Mountain J. Math. 40 (2010), no. 5, 1579--1585. doi:10.1216/RMJ-2010-40-5-1579. https://projecteuclid.org/euclid.rmjm/1289916912