非负矩阵乘积的谱半径与无穷范

Spectral Radius and Infinity Norm of the Product of Two Nonnegative Matrices

众所周知,对于矩阵A的任意范‖A‖,都有谱半径ρ(A)≤‖A‖。特别的,考虑其无穷范‖A‖∞,当ρ(A)<‖A‖∞且ρ(B)<‖B‖∞时,我们一般不能得出ρ(AB)<‖AB‖∞;但是对于两个二阶非负矩阵,本文给出了一个完整的结果,通过三个定理,完全弄清了ρ(AB)与‖AB‖∞的关系。

It is well known that the spectral radius of a matrix is less than or equal to any norm of the matrix.Forany ｜｜A｜｜ , if ρ（A） 〈 ｜｜A｜｜∞ and ρ（B） 〈 ｜｜B｜｜∞ , we can not conclude that ρ（AB） 〈 ｜｜AB｜｜∞ In this paper, we have given a clear result about nonnegative matrices in R^2×2 .