Research On Special Matrices And Matrices Inequalities

Matrix theory is not only rich in research content,but also a practical mathematical tools.The compression matrix and the positive semi-definite block matrix occupy the important status in the special matrix theory.About matrix theory plays an important role in research.The properties of the 2 × 2 positive semidefinite block matrices has always been concerned by experts and scholars,using the positive semi-definite block matrix to study matrix inequality has always been a very hot research method.In this paper,we study the properties of two special matrices in compression matrix-double random and the partial isometric matrix,and discussed the singular value inequality of 2 × 2 the positive semi-definite partitioned matrices.The main research results of this paper are divided into three chapters,the main content of each chapter is as follows:The first chapter mainly introduces the definition and lemma needed in this paper.There are mainly introduce the special matrices such as compression matrix,Partial isometric matrix,Projection matrix,Orthogonal projection matrix,Double random matrix,Nonnegative matrix,Irreducible matrices,and the definition of increasing functions and convex functions.Meanwhile,There are introduced the Hartwig-Spindelbock decomposition of the matrix,the Schur decomposition of matrix,Perron-Forbenius theorem,Courant-Fischer Minimax theorem and the correlation theorem of singular value and eigenvalue of nonnegative increasing function.In chapter two,we study two kinds of special compression matrixes:double random matrixes and partial isometric matrixes.Through the permutation decomposition of double random matrix,Schur decomposition of compression matrix and Hartwig decomposition of matrix,the related properties of double random matrix and the equivalent characterization of partial isometric matrix are studied,obtained some properties of double random matrices under certain conditions and further equivalent characterization of partial isometric matrices.It is convenient to study special matrices,matrix parallel sum,matrix partial order,and matrix lattice.In chapter three,we study the inequality of singular values of the 2 × 2 positive semi-definite block matrices.First of all,it's going to be the properties of semidefinite matrices and the eigenvalues and singular values of the positive semi-definite matrices researched the singular value inequality of the 2 × 2 positive semi-definite block matrices.About the positive semi-definite block matrices H=(?),when A,B?Mn(C),then 2?j(A*KB)? max{?A?2,?B?2}?j(H),j=1,2,…,n.Secondly,the inequalities for singular values of convex functions of semi-positive definite partitioned matrices are studied.We obtained that for positive semi-definite block matrix H=(?),if A,B ? Mn(C)such that max{?A?2,?B?2}?2 and f is a nonnegative increasing convex function on[0,+?)satisfying f(0)=0,then 2?j[f(|A*KB|)]?max{?A?2,?B?2} ?j(f(H)),the increasing convex function on the interval I.Finally,some singular value inequalities for some special semi-positive definite partitioned matrices and singular value inequalities for the special increasing convex functions are obtained.