基于Perron补的Z-矩阵最小特征值界的估计

Estimating the Bounds of the Smallest Eigenvalue of Z-matrix with Perron Complement

本文给出了估计不可约Z-矩阵的最小特征值上下界的一种简单方法,即以矩阵的广义Perron补为基础,将不可约Z-矩阵A=sI?B的最小特征值问题化为广义Perron补Ps?ρ(B)(A/Aα)的最小特征值问题,然后利用矩阵范数的性质导出了A的最小特征值界的估计式,同时也给出了非负不可约矩阵B的谱半径的一种简单估计式.

In this paper,we present a simple method to estimate the lower and upper bounds for the smallest eigenvalue of irreducible Z-matrices,the method is based on the generalized Perron complement.For the smallest eigenvalue problem of the irreducible Z-matrix A = sI - B,we convert it into the smallest eigenvalue problem of a generalized Perron complement.Then we utilize the properties of matrix norms and obtain the estimation of the bounds for the smallest eigenvalue of A.Moreover,we give a simple estimation for the spectral radius of a nonnegative irreducible matrix.