Two inequalities involving Hadamard products of positive semi-definite Hermitian matrices

Abstract

We extend two inequalities involving Hadamard products of positive definite Hermitian matrices to positive semi-definite Hermitian matrices. Simultaneously, we also show the sufficient conditions for equalities to hold. Moreover, some other matrix inequalities are also obtained. Our results and methods are different from those which are obtained by S. Liu in [J. Math. Anal. Appl. 243:458-463(2000)] and B.-Y. Wang et al in [Lin. Alg. Appl. 302-303:163-172(1999)].