Semidefiniteness without real symmetry

Charles R. Johnson, Robert Reams
2000 Linear Algebra and its Applications  
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
more » ... 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).
doi:10.1016/s0024-3795(99)00256-6 fatcat:bdlmcw3vybcmhmsowq4rxezt6q