The necessary and sufficient conditions for the matrix equation AXB=D having symmetric and semipositive subdefinite solutions are studied.

The necessary and sufficient conditions for thd matrix equation (XA,XB)=(C,D) having symmetric, semi-definite and semipositive subdefinite solutions are studied.

In this paper,the sufficient and necessary condition for the consistency on the normal subpositive definite matrix solutions of the matrix equation X+A TX -1 A=I is derived,thus the sufficient and necessary condition for the consistency of it s inverse problem is obtained,and its general solutions are given,as well as an estimation of spectral radius about its solutions.

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.

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.

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.