Inequalites Of Matrix Special Product

This paper investigates a matrix inequality on the kronecker product that unifies the proofs of many existing inequalites in the Lowner partial ordering on the sum, ordinary product and Hadamard (Schur) product in section 1. for example, the famous Haynsworth's inequality can be obtained from our generic inequality. Our merits in yielding a variety or inequalites by directly choosing different numbers and matrics in the inequality. Schur complements serves as the basic tool, then we use the same way to get some matrix inequalities in the Moore-Penrose inverse in the section 2. For partitioned matrices, Khatri-Rao and Tracy-Singh product can be seen the generalized Hadamard and kronecker product each other, Rao, kleff and liu has recently got some inequalites of partitioned matrices involved khatri-Rao product. The kind of product is more useful in estimation and hypothesis testing in Linear time series regressions with unit roots and limited information maximum livelihood estimation for analysis of survely data. We gives another proofs of these inequalities involving khatri-Rao product of partitioned rnatris in section 3 and we get some inequalites of majorization of the Hadamard product.