A Represemtation Theotem For (trA～p)～(1/p) And The Inequality On Trace And Eigenvalues Of The Schur Complement Of Positive Definite Hermitian Matrix

The paper is a research on the representation of positive semidefinite matrix and the relationship between the trace and eigenvalues between the Schur complement of the sum of positive definite Hermiteian matrix and the sum of the Schur complements of positive definite Hermiteian matrix.It consists of the following two parts.The first part:Under the conditions 0 < p < 1 and p < 0, we obtain the inequality:tr AX ≥ (trA~p)~1/p(A is a positive semidefinite matrix), and get a representation for (trA~p)~1/p:From it we follow matrix versions of the inequalities of Holder and Minkowski as their application.The second part: We get inequalities about the trace and eigenvalues between the Schur complement of the sum of positive definite Hermiteian matrix and the sum of the Schur complements of positive definite Hermiteian matrix.