Communications in Mathematical Physics

, Volume 254, Issue 3, pp 651–658 | Cite as

On Some Mean Matrix Inequalites of Dynamical Interest

  • Igor RivinEmail author


Let ASL(n,ℝ). We show that for all n>2 there exist dimensional strictly positive constants C n such that

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where ||A|| denotes the operator norm of A (which equals the largest singular value of A), ρ denotes the spectral radius, and the integral is with respect to the Haar measure on O n , normalized to be a probability measure. 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. This answers questions asked in [3, 5, 4]. We also discuss what happens when the integral above is taken with respect to measure other than the Haar measure.


Neural Network Statistical Physic Complex System Positive Constant Probability Measure 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsTemple UniversityPhiladelphia

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