The Expressions For The Generalized Inverses Of Block Matrices

We all know that if M is a square non-singular matrix, there exists a matrix G such that MG = GM =I, which is called the inverse of M and denoted by M^-1. If M is a singular or a rectangular matrix, no such matrix G exists. E.H.Moore and R.Penrose extended the notion of inverse and raise the definition of generalized inverse .There are all sorts of generalized inverses , among all Moore-Penrose inverse, except that the M-P inverse M^? is unique, the others are not unique. Additionally, several important generalized inverses, including the M-P inverse M^? , the weighted M-P inverse M_X,Y^?, the group inverse M_g and the Drain inverse M_d are actually some kinds of {2} inverse M_T,S² with corresponding prescribed ranges and null spaces . They have broad applications in linear equations,differential equations,difference equations,optimal control and so on.In this paper, we study the representations of the generalized inverses, and the size or difficulty of these problems may be reduced if M is partitioned into its blocks.We can compute things about M from the submatrices. For an arbitrary matrix M∈C_r^m×n, through the rearrangement of rows and columns, we can bring one nonsingular r×r submatrix to the upper left hand corner of M, that is to say that there exist permutation matrices P and Q such thatBased on it, we can obtain all kinds of expressions of generalized inverses. If we only per-form the permutation of rows or the permutation of columns, we have M = P (?) orM = (?)Q, where A are full row rank and full column rank matrices with rank r respectively.Thus we get the outcomes corresponding to the four block matrix.Chapter 1 of this paper explains the notations used in the paper and several definitions of generalized inverses.The expressions of M-P inverses,M_T,S² and its some special cases are derived when M is partitioned into four blocks and two blocks by multiplying the permutation matrices in chapter 2 and chapter 3 respectively , what's more, we give some numerical examples to illustrate the former theories, they show that our results are effective.