Semidefiniteness without real symmetry

✍ Scribed by Charles R. Johnson; Robert Reams

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 71 KB
• Volume: 306
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00256-6

✦ Synopsis

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 (k 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).