# On Some Mean Matrix Inequalites of Dynamical Interest

@article{Rivin2003OnSM,
title={On Some Mean Matrix Inequalites of Dynamical Interest},
author={Igor Rivin},
journal={Communications in Mathematical Physics},
year={2003},
volume={254},
pages={651-658}
}
• Igor Rivin
• Published 2 December 2003
• Mathematics
• Communications in Mathematical Physics
AbstractLet A ∈ SL(n,ℝ). We show that for all n>2 there exist dimensional strictly positive constants Cn such that 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 On, normalized to be a probability measure. The same result (with essentially the same proof) holds for the unitary group Un in place of the orthogonal group. The result does not hold in dimension 2…
4 Citations
GL
. We consider orthogonally invariant probability measures on GL n ( R ) and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix
Random and mean Lyapunov exponents for $\mathrm{GL}_n(\mathbb{R})$
• Mathematics
• 2022
. We consider orthogonally invariant probability measures on GL n ( R ) and compare the mean of the logs of the moduli of eigenvalues of the matrices to the Lyapunov exponents of random matrix
Estimates and identities for the average distortion of a linear transformation
For a linear transformation A from Rn to Rn, we give sharp bounds for the average distortion of A, that is, the average value of log of the euclidean norm of Au over all unit vectors u. This is
1. Geometry of polyhedra and related subjects
In my doctoral dissertation (directed by W. P. Thurston) I studied the geometry of convex polyhedra in hyperbolic 3-space H3, and succeeded in producing a geometric characterization of dihedral

## References

SHOWING 1-8 OF 8 REFERENCES
Random Versus Deterministic Exponents in a Rich Family of Diffeomorphisms
• Mathematics
• 2002
We study, both numerically and theoretically, the relationship between the random Lyapunov exponent of a family of area preserving diffeomorphisms of the 2-sphere and the mean of the Lyapunov
A formula with some applications to the theory of Lyapunov exponents
• Mathematics
• 2002
We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub,
Regions in the Complex Plane Containing the Eigenvalues of a Matrix
• Mathematics
• 1994
The Gersgorin circle theorem gives a region in the complex plane which contains all the eigenvalues of a square complex matrix. It is one of those rare instances of a theorem which is elegant and
Matrix analysis
• Mathematics
Statistical Inference for Engineers and Data Scientists
• 2018
This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.
Polya. Inequalities
• 1988
On random and mean exponents for unitarily invariant probability measures on GL(n,C)
• Technical report,
• 2001