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On some mean matrix inequalities of dynamical interest

Igor Rivin
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Abstract

Let 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.

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Igor Rivin. 2003-12-11. On some mean matrix inequalities of dynamical interest. https://arxiv.org/abs/math/0312048

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