Research On The Z-matrix And M-matrix Generalized Perron Complement Matrix

The concept of the Perron complement of a nonnegative and irreducible matrix was introduced by Meyer in 1989 and it was used to construct an algorithm for computing the stationary distribution vector for markov chains. He give lots of properties of it, the concept of the generalized Perron complement of the nonnegative matrix is introduced because it is useful to estimate the Perron root of nonnegative matrix. Many famous mathematician do lots of investigation of it since there are lots of interest and useful properties of the generalized Perron complement of the nonnegative matrix. It has been extended to the inverse M - matrix and totally positive matrix. Here we mainly discuss the generalized Perron complements of the irreducible M - matrix and Z - matrix.This paper mainly includes two parts:1. The Perron complement of M-matrix: We discuss when the generalized Perron complement of M-matrix is still M-matrix and some other properties of it. For example:In chapter two, we give thatLet K = sI - M be an irreducible M - matrix withρ( M )+ t≤s. Then the generalized Perron complement P_t ( K / K [α]) is also an irreducible M - matrix.Let K = sI - M be an irreducible M - matrix withρ( M )+ t≤s. Then ,q ( P_t ( K / K [α])) is a strict decreasing function of t on ( -∞, s -ρ( M)].Let K = sI - M be an irreducible M - matrix withρ( M )+ t≤s. Then ,q ( P_t ( K / K [α]))≥q ( K), the equality holds forρ( M )+ t = s.Let K = sI - M be an irreducible M - matrix withρ( M )+ t≤s. Then, for the three M - matrices K [β], Pt ( K / K [α]), - ( K / K[α]), the following results hold:1) K [β] > - ( K / K [α])≥P_t ( K / K [α]) for 0 < t≤s -ρ( M)2) K [β] > P_t ( K / K [α]) = - ( K / K [α]) for t= 03) K [β] > P_t ( K / K [α])≥- ( K / K [α]) for t< 02. The generalized Perron complement of Z - matrix: We prove that the generalized Perron complement of Z - matrix is still the Z - matrix. Comparing the discussing of the second chapter we give the similar results in third chapter. For example:Let K = tI - M be an irreducible Z - matrix, then the generalized Perron complement P_x ( K / K [α]) is an irreducible Z - matrix for x < -ρ( M [α])+ t.Let K = tI - M be an irreducible Z - matrix, then n ( P_x ( K / K [α])) is an strictly decrease function of x for x < -ρ( M [α])+ t.etc.