The Representations For The Drazin Inverse Of The Sum Of Two Matrices And The Block Matrix

The Drazin inverse of a square matrix has various applications in singular equations and singular difference equations, operator theory, Markov chains, cryptogram, and iterative methods. Therefore, Drazin inverse has been a very important research area since the middle of the last century. Up to now it is still one of extremely active research branches on international.In 1958, M. P. Drazin presented the generalized inverse based on the integration of ring and semi group, which was later called Drazin inverse. First, we consider the Drazin inverse ( P + Q)D of the sum of two matrices P and Q , this problem was first considered by Drazin in 1958 in his celebrated paper. The cases PQ = 0, P 2 Q = 0 and PQ 2 = 0 had been studied, our study is based on the known result, and generalizes the known results.In 1979, Campbell and Meyer first proposed the problem of finding a formula for the Drazin inverse of in terms of the blocks of the partition, where A and D are square complex matrices but need not to be the same size. This problem is quite complicated. To the best of our knowledge, there was no representation for the Drazin inverse of M with arbitrary blocks published. In chapter 3, according to split M among the sum of two matrices, then using the formula of the Drazin inverse for the sum of two matrices, finally, we obtain the repreaentation of the Drazin inverse of M .Recent years, N.Castro-González et al. studied the Drazin inverse of the block matrix such as , where A is a square matirx, and obtained some important results. Based on these results, in chapter 4, we consider the Drazin inverse of M when A is a idempotent.As we know, group inverse is a special Drazin inverse, but it doesn't exist for every square matrix. So, it is great necessary to study the existence of the group inverse , especially for the block matrix. In chapter 5, when A is a idempotent, we give the necessary ang sufficient conditions for the existence and the representation of the group inverse for block matrix...