Address. Xian Zhang, Department of Mathematics, Heilongjiang University, Harbin, 150080, P.R.C.
School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast,
Ashby Building, Belfast, BT9 5AH, Northern Ireland, United Kingdom
Zhong-Peng Yang, Department of Mathematics, Putian College, Putian, Fujian, 351100, P.R.C.
Chong-Guang Cao, Department of Mathematics, Heilongjiang University, Harbin, 150080, P.R.C.
Abstract. We extend three inequalities involving
the Hadamard product
in three ways. First, the results are extended to any partitioned blocks
Hermitian matrices. Second, the Hadamard product is replaced by
the Khatri-Rao product. Third, the necessary and sufficient conditions under which equalities
occur are presented. Thereby, we generalize two inequalities involving the
AMSclassification. 15A45, 15A69.
Keywords. Matrix inequality, Khatri-Rao product, Tracy-Singh
product, Hadamard product,
Kronecker product, Schur complement.