The Research On The Upper Bound Of The Index Of The Matrix

Let Cm×n be the set of all m×n matrices over the complex number field. For A∈Cn×n, the smallest nonnegative integer k such that rank(Ak)= rank(Ak+1) is called the index of A, denoted by Ind(A). For A∈Cn×n, the unique matrix X∈Cn×n satisfying the following equations AkXA=Ak,XAX=X, AX=XA is called the Drazin inverse of A, denoted by AD, where k=Ind(A). Particularly, when Ind(A)≤1, AD is called the group inverse of A, denoted by A#. Furthermore, the smalle-st smallest nonnegative integer k such that the group inverse of Ak exists is the index of matrix A, and AD= Ak-1(Ak)#=(Ak)# Ak-1The Drazin(group) inverses of block matrices have important applications in singular differential equations and singular difference equations, Markov chains, iterative methods, cryptography and so on. The research on the Drazin inverses for block matrices is mainly the research on the representations and the indices for the Drazin inverses for matrices, and the indices have important significance in the computations and representations for the Drazin inverses for block matrices.This paper introduces the research background, the research status, the research significance of this subject in Chapter 1, and the basic knowledge related to this paper in Chapter 2. Next, the main results of this paper are given in Chapter 3 and Chapter 4 respectively, which are listed below:1. The bounds of the index of block matrix ((?)(A, C are the matrices of the same order) are given, where B satisfies the following conditions:(1) B= c1A+c2C, where c1, c2 are complex numbers and c1+c2≠0;(2) B=ApCq where p, q are positive integers and p< Ind(A), q<Ind(C);(3) AmBCn= 0, where m≤Ind(A), n≤Ind(C), and this paper also gives the bounds of the index of the block matrix(?), where BC＝0.2. The upper bound of the index of the matrix P+Q is given, where PQ=0 and P,Q∈Cn×n, and applies the conclusion, we give the condition of the existence and the representations of the group inverse of the matrix P+Q. With the representation, this paper also gives the condition of the existence and the representations of the group inverses of block matrix(?) under the assumption of CA=0, CBD=0.