Positive Semidefinite Matrices

Abstract

This chapter studies the positive semidefinite matrices, concentrating primarily on the inequalities of this type of matrix. The main goal is to present the fundamental results and show some often-used techniques. Section 6.1 gives the basic properties, Section 6.2 treats the Löwner partial ordering of positive semidefinite matrices, and Section 6.3 presents some inequalities of principal submatrices. Section 6.4 derives inequalities of partitioned positive semidefinite matrices using Schur complements, while Sections 6.5 and 6.6 investigate the Hadamard product and Kronecker product and related matrix inequalities. Finally, Section 6.7 shows matrix inequalities of the Cauchy-Schwarz type.