Furthermore, a new estimation of the lower bound of the determinant module on the Hadamard product of a Hermitian positive definite matrix and a quasi-generalized complex positive definite matrix is obtained by using the improvement and the properties of quasi-generalized complex positive definite matrices.

Theorefore, we obtain the necessary and sufficient conditions under which the relative gain array of Hermitian positive definite matrix becomes identity matrix.

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.