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 7.1 gives the basic properties, Section 7.2 treats the L¨owner partial ordering of positive semidefinite matrices, and Section 7.3 presents some inequalities of principal submatrices. Section 7.4 derives inequalities of partitioned positive semidefinite matrices using Schur complements, and Section 7.5 and 7.6 investigate the Hadamard product of the positive semidefinite matrices. Finally, Section 7.7 shows the Cauchy–Schwarz type matrix inequalities and the Wielandt and Kantorovich inequalities.