Matrix Inequalities On Generalized Schur Complement

With the fast development of technology, matrix theory, for its vast utility, has developed into a powerful tool in solving many problems in finite dimensional space. While the study developing gradually, scholars have been introducing more and more new methods. This gives birth to Schur complement and Perron complement, which now become two important tools in studying matrices. Matrix inequality has been an important branch of matrix theory for a long time. In recent years, many authors focus on inequalities involing Schur complement and Perron complement, and there have been many results in the literature. In this paper, we introduce the concepts of generalized Schur complement and generalized Perron complement, and obtain many important inequalties involing them. We devide our investigation into three chapters:Chapter 1 works on some Lowner partial ordering of positive semidefinite matrices and their generalized Schur complements of which a minimal pricipal is offered. We prove the following: (1) If , then for an arbitrary , where ; (2) Let , . Then and , furthermore, ; (3) and equality holds if and only if there exits some permutation matrix P, such that ; (4) If min{m,n), and , then ; (5) (BAB)/a and can not be compared in general even for A, B > 0. But if , satisfying , then . In the second part, we introduce the concept of Kronecker product and show that if , , and , then , where . In the thid part, some inequalities involving Hadamard product are given: (1) Let A, B be positive definite, r be an arbitrary positive integer. Then ; ; and . (2) For , it can be shown that , and In chapter 2, using some well-known results, several inequalities on eigenvalues of the generalized Schur complement of BAB* where A is positive semidefinite are obtained: (1) If , , , and l is an positive integer satisfying 1 < I < n - r. then . (2) Let A be positive semidefinite, . Then .Chapter 3 deals. with quotient property and the eigenvalues interlacing results of generalized Schur complements of semidefinite matrices, as well as of their inverse, Moore-Penrose inverse. By offering some counter-examples, we show that for an arbitrary Hermitian matrix H, these two theorems are not necessarily true after proving that they do hold when H is semidefinite. We prove that if H is n x n semidefinite matrix, then the eigenvalues of HjA interlace those of H. Furthermore, if A is an r x r pricinpal submatrix of H, A11 is an r1 x r1 pricipal submatrix of A, then A/A11 is a principal submatrix of H/A11 and H/A = (H/A11)/(A/A11). Thus we give a relatively complete answer to the question whether the quotient property of Schur complement can be extended to the generalized Schur complement as some other properties can do. Then we introduce the concept of the generalized Perron complement for m x n complex matrices and derive similar quotient property and eigenvalues interlacing results for symmetry nonnegative matrices and positive semidefinite matrices. Let a C N = {1,2, ...,n}, , . We have the following: (1) If is semidefinite. then if A is positive semidefinite; if A is negative semidefinite where 1 < i < n - r; (2) If is semidefinite, then ; (3) If A is n x n positive semidefinite, , then for . (4) Let A be an arbitrary symmetric nonnegative matrix of order n, and . Then is a principal submatrix of , and , for arbitrary .