A simple proof for the inequality between the perron root of a nonnegative matrix and that of its geometric symmetrization

Let A = (aij) be a nonnegative square matrix, let G = (gij) be its geometric symmetrization, i. e., gij =→aijaji, and let ρ denote the Perron root. We present a simple proof for the well-known inequality ρ(A) ≥ ρ(G). © 2010 Pleiades Publishing, Ltd.