Inequalities Involving Khatri-Rao Products of Positive Semi-definite Matrices ∗

Several inequalities involving the Khatri-Rao products of two four-block pos- itive definite real symmetric matrices are established by Liu in (1). We extend these to two general positive semi-definite real symmetric block matrices and nec- essary and sufficient conditions under which these inequalities become equalities are presented. Let S(m) be the set of all real symmetric matrices of order m. Consider matrices M ∈ S(m )a ndN ∈ S(p) which are partitioned as follows