非负不可约矩阵谱半径的估计

兰州理工大学学报 ›› 2025, Vol. 51 ›› Issue (5): 163-166.

非负不可约矩阵谱半径的估计

李娜¹, 钟琴^*2

1.吉利学院智能科技学院, 四川成都 641423;
2.四川大学锦江学院数学教学部, 四川眉山 620860

收稿日期:2023-09-26 出版日期:2025-10-28 发布日期:2025-10-25
通讯作者: 钟琴(1982-),女,四川自贡人,教授. Email:bbs3_zq@126.com
基金资助:
眉山市科技厅项目(2024KJZD163)

Bounds for the spectral radius of nonnegative irreducible matrices

LI Na¹, ZHONG Qin²

1. School of Intelligence Technology, Geely University of China, Chengdu 641423, China;
2. Department of Mathematics, Sichuan University Jinjiang College, Meishan 620860, China

Received:2023-09-26 Online:2025-10-28 Published:2025-10-25

摘要/Abstract

摘要： 借助非负矩阵的非零行和, 将经典的Gerschgorin圆盘定理和Hölder不等式应用于非负矩阵谱半径的估计,得到非负矩阵谱半径易于计算的上下界表达式.该方法可操作性强,且能得到较紧的上下界.通过具体的算例验证结果的精确性.

关键词: 非负矩阵, 不可约, 谱半径, 上界, 下界

Abstract: Based on the nonzero row sums of nonnegative matrices, the classical Gerschgorin disk theorem and Hölder inequality are applied to estimate the spectral radius of a nonnegative matrix, and the expressions of the upper and lower bounds of the spectral radius of a nonnegative matrix are obtained. The proposed methods have good operability and can get tight upper and lower bounds. Finally, the accuracy of the results is verified by specific arithmetic examples.

Key words: nonnegative matrix, irreducible, spectral radius, upper bound, lower bound

中图分类号:

O151.21

李娜, 钟琴. 非负不可约矩阵谱半径的估计[J]. 兰州理工大学学报, 2025, 51(5): 163-166.

LI Na, ZHONG Qin. Bounds for the spectral radius of nonnegative irreducible matrices[J]. Journal of Lanzhou University of Technology, 2025, 51(5): 163-166.

参考文献

[1] PERRON O.Zur theorie der matrices [J].Mathematische Annalen,1907,64:248-263.
[2] ANDRADE E,CIARDO L,DAHL G.Perron values and classes of trees [J].Linear Algebra andIts Applications,2022,639:135-158.
[3] CHERIYATH H,AGARWAL N.On the Perron root and eigenvectors associated with a sub-shift of finite type [J].Linear Algebra and Its Applications,2022,633:42-70.
[4] 钟琴,赵春燕,王妍,等.非负矩阵最大特征值的上下界估计 [J].高校应用数学学报(A辑),2022,37(4):484-490.
[5] ADAM M,ARETAKI A.Bounds for the spectral radius of nonnegative matrices and generalized Fibonacci matrices [J].Special Matrices,2022,10(1):308-326.
[6] ADAM M,AGGELI D,ARETAKI A.Some new bounds on the spectral radius of nonnegative matrices [J].AIMS Mathematics,2019,5(1):701-716.
[7] LIAO P.Bounds for the Perron root of positive matrices [J].Linear and Multilinear Algebra,2022,71(11):1849-1857.
[8] 王信存,吕洪斌.非负矩阵最大特征值的拟幂型算法 [J].东北师大学报(自然科学版),2022,54(4):6-11.
[9] WANG G,LIU J.An improved diagonal transformation algorithm for the maximum eigenvalue of zero symmetric nonnegative matrices [J].Symmetry,2022,14(8):1707.
[10] LEDERMANN W.Bounds for the greatest latent roots of a positive matrix [J].Journal of the London Mathematical Society,1950(4):265-268.
[11] OSTROWSKI A.Bounds for the greatest latent root of a positive matrix [J].Journal of the London Mathematical Society,1952(2):253-256.
[12] BRAUER A.The theorems of Ledermann and Ostrowski on positive matrices [J].Duke Mathematical Journal,1957,24(1):265-274.
[13] CAO D S.Bounds on eigenvalues and chromatic numbers [J].Linear Algebra and Its Applications,1998,270(1/2/3):1-13.
[14] BRAUER A,GENTRY I C.Bounds for the greatest characteristic root of an irreducible nonnegative matrix [J].Linear Algebra and Its Applications,1974,8(2):105-107.
[15] 逄明贤.矩阵谱论 [M].长春:吉林大学出版社,1989.