The definition of the positive hyperstable matrix was given,and the relations among classes of positive definite matrix,metapositive definite matrix, well generalized positive definite matrix,generalized positive definite matrix,positive hyperstable matrix and positive stable matrix were studied.

We finded differently out the Complex congruence canonical forms of metapositive definite matri ces and metapositive semidefinite matrices,and obtain a necessary and sufficient condition about two metapositive definite matrix being Complex Congruen

Let A be an n×n metapositive definite matrix.

Two sufficient conditions of subpositive definite matrices were established,and some applications in devotion output and control theory were given.

Thispaper studiesthe inequality theory of subpositive definite matrices,gainssome new propertiesof it,enrichesitstheory and widensthe range of itsapplication.

In the remark, the mistakes of a series of results in concerned with sub-positive definite matrix are proved by twoexamples.

Kuo, and we also give a necessary and sufficient condition such that the tensor product of two semipositive subdefinite matrices is a semipositive subdefinite matrix too.

Recently, however, attentions are been paying to the situation for S to be the set of general(semi) positive definite matrices(called as semipositive subdefinite matrices below) .

A Minkowski inequality over matapositive definite matrix is proved,and the results of some references are generalized and improved.