Research On The Solution Of A Class Of The Descriptor System With Drazin Inverse

This paper studies the applications of Drazin (group) inverse of block matrices in the representations of the solutions of a class second order system, and give the following results.1. We give the representations for the Drazin inverse for (?) (E and F are square) under the condition EF= FE. So we solve the problem partially that to find an explicit representation of the Drazin inverse for the block matrix (?) (E and F are coefficient matrices in singular differential equations) which proposed by Campbell in 1983. And we also give the Drazin inverse for (?) with E square under the condition EFG= FGE.2. We study the explicit representations for the Drazin inverse of M=(?) in terms of A,B,C,D,AD,(BC)D and DD under the conditions ABC=0 and either DC=0 or BD=0. These results generalize the main results of [6] and [11].3. We study the existence and representation of the group inverse of block matrix whose subblock is the linear combination of matrices or the product of matrices. We give the necessary and sufficient conditions for the existence of the group inverses of block matrix over skew field and the corresponding representations under the following conditions: (1) B# and C# exist, A=c1B+c2C, nonzero elements c1,c2 are the center of the skew field; (2)A=BkCl, k and l are positive integers.4. We give the necessary and sufficient conditions for the existence and the representations of the group inverse of block matrix (?) over skew field and the corresponding representations under the following conditions:(1) B# and (BÏ€A)# exist;(2) B# and (ABÏ€)# exist; (3)B# exists and BABÏ€=0ï¼›(4)B# exists and BÏ€AB=0.5.We study the existence and the representations of the group inverse for two class block matrices with an inversible subblock over the skew field and give the corresponding representations.One is the block matrix (?),where A is square and one of A,B,C is an invertible matrix.The other is the block matrix (?),where A is square and one of A,B,C and D is an invertible matrix.