Some important inequalities of block P-Schur complement and Khatri-Rao product for positive semidefinite block matrices were studied in this paper,which generalized some results.

We first discuss the connections between Euclidian distance matrix and positive semidefinite matrix under the condition that Ax 0=λx 0, λ≥0, x 0=en, A n×n is a positive semidefinite matrix.

This paper is concerned with the problem of real symmetric positive semidefinite matrix pencil under spectral restriction.

There exist great differences between positive semidefinite matrix and positive definite matrixin the inequality research.

Applying these results, the inequality of Khatri-Rao product about positive semi-definite matrices is generalized to real symmetric matrices, and its inverse inequality and equational condition are also given.

Furthermore,theorem 2 gives and proves a suficient and necessary conditionan for the case of positive semi-definite matrices B by the method of matrices decomposition and block matrice.

It shows the proof of four points on a circle by the knowledge of determinant;the methods of resolving applied problems by theories about the solution of homogeneous linear equations;and the proof of inequality by positive definite and positive semi-definite matrix.

In this paper, two inequalities of the positive semi-definite matrix trace are given.