Some Properties Of Partitioned Positive Semi-definite Matrix And Special Matrix Function

Partitioned positive semi-definite matrix is an important part of matrix theory The study and use of the properties of partitioned positive semi-definite matrix have always been the hot research contents.Matrix function that is widely used in the actual life is another important research branch of matrix theory.Based on the properties of partitioned positive semi-definite matrix and matrix function studied by experts,this paper mainly further study the problem of maintaining semi-definiteness of partitioned positive semi-definite matrix with respect to matrix index,the vivid description of matrix Schur complement and the basic properties and relationships of some special matrix function and got some important propert iesThe main contents of this paper can be summarized as followsThe first chapter mainly introduces some basic symbols,concepts,and some theorems used in this paper.For example,the definitions of positive matrix,positive semi-definite matrix,normal matrix,Schur complementary matrix,the product of Harmard,the inner product of Frobenius and the theorems of the judgment theorem of positive semi-definite matrix and Schur complementary theoremIn chapter two,based on the properties of partitioned positive semi-definite matrix that have been studied by the experts.Firstly,this paper study the two by two partitioned positive semi-definite matrix,which is taken the spectral ra-dius,spectrum span for every block matrix respectively,the resulting matrix is still positive semi-definite;Secondly,the equivalent conditions of partitioned positive semi-definite matrix that,is taken a special block are studied and some basic prop-erties of partitioned positive semi-definite matrix are obtained;Finally,using the properties of the matrix Schur complement,the characterization and the properties of two by two matrix Schur complement are obtainedIn this chapter,by using the definitions and properties of sesquilinear func-tion,Frobenius norm and commutator,some basic properties of covariance matrix function,variance matrix function,wigner-yanase skew information,wigner-yanase correlation and other matrix functions are obtained,besides the relationships among them are also obtained.