Eigenvalue Estimate Of M-matrix And Spectrum Radius Properties Of Nonnegative Matrix

The properties and the smallest eigenvalue of M-matrix are studied.The eigen-values of the Hadamard product and the Fan produat of matrices are estimated and compared with conclusions of Literiture[2].The spectrum radius properties of nonnegative matrix are characterized.It is divided into 4 chapters in this paper. Some main results are lists as follows.In chapter l,some notations are introduced and some well-known theorems are given. In section two,we introduce the definitions of M-matrix;Fan prod-uct;Hadamard product;stochastic matrix;spectrum radius and so on.In section 3,we give some well-known theorems.In chapter 2,Some properties of M-matrix and the smallest eigenvalue of M-matrix are proved by using equivalent definitions and theorems of M-matrix.In chapter 3,Hadamard product and Fan product of lower bound of minimal eigenvalue problem for matrix are discussed.The following five results are proved.Let (?)n be the set of all n×n nonsingular M-matrices.Firstly,if A=(aij)∈(?)n,B= (bij)∈(?)n,B-1=(βij),then T(A o B-1)≥(?)[2aiiβii-T(A)βii+(T(A)-aii)/(T(B)];Sec-ondly,if A=(aij)∈(?)n,B=(bij)∈(?)n,B-1=(βij),then T(A o B-1)≥T(A)T(B)(?){((aij)/T(A)+(bij)/T(B))-1)(βij)/bii}；Thirdly,if A=(aij),B=(bij)∈Rn×n∈(?)n,and B is irreducible.Suppose x=(xi),y=(yi)∈Rn are right and left Perron eigenvectors of B-1 define t=x o y,then T(A.B-1)≥(T(A))/(T(B))(min{t1,t2,…,tn})/t1,+t2…+tn);Fourthly ,if A=(aij)∈Rn×n∈(?)n,B=A-1≡(βij)∈Rn×n.Suppose x=(xi)∈Rn is right Perron eigenvector of B,then T(A.A-1)≥1+(?){((1-aii)xm)/(aiixi)),includ-ing xm=max{x1,x2,…,xn)；Fifthly.if A=(aij)∈(?)n,B=(bij)∈(?)n,then丁(AB)≥(?)[aiiT(B)+biiT(A)-T(A)T(B)].At the same time,some results are compared with corresponding conclusions of Literiture[2].In chapter 4,some properties of nonnegative matrix are discussed and some simple conclusions are given.