2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69428Let A be an n by n matrix with determinant 1. We show that for all n > 2 there exist dimensional strictly positive constants C_n such that the average over the orthogonal group of log rho(A X) d X > C_n log ||A||, where ||A|| denotes the operator norm of A (which equals the largest singular value of A), rho denotes the spectral radius, and the integral is with respect to the Haar measure on O_n The same result (with essentially the same proof) holds for the unitary group U_n in place of the orthogonal group. The result does not hold in dimension 2. We also give a simple proof that the average value over the unit sphere of log ||A u|| is nonnegative, and vanishes only when A is orthogonal.11 pages; revision shows notes that it is essentially necessary to use Haar measure (class)Dynamical SystemsFunctional Analysis37D25;37A25; 15A45; 15A52text