| Matrix theory has very rich contents as an important branch of mathematics, it provides a useful tool for mathematics and other scientific fields. In the study of the theory of the matrix, the canonical form problem has important position and been widely used in mechanics, control theory, system analysis, and other fields, so it is significant to discuss the Jordan canonical form of special block matrix. Generalized inverse is the essential part in matrix theory, especially some equality and inequality about the generalized inverse have aroused great interest of most researchers, thus it is necessary to study on the generalized inverse of matrix equation A-+B-=A-(A+b)b-In this paper, we discuss the Jordan canonical forms of four kinds of special block matrixes, and simplify the proof of a theorem are given by applying the results. In addition, we investigate the equivalent problems associated with the matrix equation A-+B-=A-(A+B)B-The first chapter, we firstly introduce some mathematical symbols and reviewed the basic concepts of Jordan block, Jordan canonical form, the k-th determinant factor of n-th matrix referred in this article. Then we present conceptions of the Kronecker product of two matrixes and generalized inverse matrix of a matrix. Then, we introduce several important theorems in Matrix theory.The second chapter, we obtaine the Jordan canonical forms of four kinds of special block matrixes by mathematical induction and simplify the proof about the number and size of the Jordan block associated with an eigenvalue of the Kronecker product of two matrixes. The third chapter, we firstly provide a simpler structured proof about the important theorem in reference [12], and propose an equivalence theorem and two corollaries about the generalized inverse of matrix equation A-+B-=A-(A+B)B-... |
PDF Full Text Request