The Representation Of Drazin Inverse For Block Matrices And The Application Of Generalized Inverse In Matrix Equation

Generalized Inverse is widely used in many areas such as differential equation, integral equation, operator theory, statistics, cybernetics, Markov chain, optimization and so on so forth. Therefore, Generalized Inverse has been a very important research area since the middle of the last century. Up to now it is still was one of extremely active research branches on international.In 1958, M.P.Drazin presented generalized inverse based on the integration of ring and semi group , which was later called Drazin inverse. In 1980, Cline and Gre-ville proposed the concept of weighted Drazin inverse based on rectangle matrix. It was the extention of square Drazin inverse and was actually quite applicable. In the past decades some experts have been devoting themselves to the research of block matrix Drazin inverse and all the researches were carried out on condition that each block was restricted. All the research fruit simply found partial solutions to the problems given by Campbell in 1983. In this essay, Chapter 1 and 2 mainly deal with forms of Drazin inverse of block matrix 2×2. Taking advantage of polynomial expansion of Drazin inverse of two matrixes sum and existing expressions of Drazin inverse of block matrix the essay provides conclusions of representation of Drazin inverse , which are additions to existing findings to some degrees. Besides, in chapter 2 it gives a detailed analysis of rectangle block matrix, which completes the structure of this essay.Apart from the discussion on forms of Drazin inverse of block matrix, this essay further explores the application of generalized inverse to rhetorical question of matrix equation. Through the decomposition of matrix, turning optimal approximation of solution with special structure into least-squares problem of linear equations is a meaningful supplement in the study of matrix equation. At last, we also give the positive solution to some matrix equations.