A Simple Proof for the Inequality Between the Perron Root of a Nonnegative Matrix and that of Its Geometric Symmetrization

摘要

Let A = (a_(ij)) be a nonnegative square matrix, let G = (g_(ij)) be its geometric sym metrization, i.e., (g_(i,j)=a_(ij)a_(ji))/(1/2), and let p denote the Perron root. We present a simple proof for the well-known inequality ρ(A) ≥ ρ(G).