非负矩阵谱半径估计的研究

摘 要

本文目标为讨论非负矩阵谱半径估计1类方法。在盖尔圆盘定理及Frobenius界值定理基础上，对这类方法给出不同程度的改进，使新界值更精确。

利用Perron补的概念，提出非负不可约矩阵谱半径界值的1个新的估计算法。该算法利用Perron补保持原矩阵的非负不可约性及谱半径的性质，使新得到的矩阵最大行和变小，最小行和变大，从而得到比Frobenius界值定理更精确的界。详细论述算法思想并给予严格证明。给出适当的数值例子，比较新算法相对于Frobenius界值定理的改进效果，最后简要评价各算法，并讨论矩阵特征问题的研究方法。

关键词 非负矩阵；谱半径；界；估计；Perron补

Abstract

This paper focuses on discussion of a class of estimation methods for spectral radius of nonnegative Matrix.based on Gerschgorin Disk theory and Frobenius’theory,these methods improve the former theories and provide sharper bounds.

Furthermore,the concept of Perron complement is introduced a new estimating method for spectral radius of nonnegative irreducible matrix is proposed and explained in detail.A new matrix dereved preserves the spectral radius while its minimun row sum increases and its minimun row sum decreases.Detail designing method and strict proof are provided with illustration of numerical examples.Finally,these algorithms’characters and the studying methods for matrix eigenproblems are also briefly discussed.

Keywords nonnegative Matrix；spectral radius；bounds；estimation；Perron complement