On the bounds of the entropy defined for an orthogonal matrix

Abstract

The optimization bounds for the Shannon's entropy of an orthogonal matrix have been found for the various classes of orthogonal matrices. A universal upper bound on the entropy for all n has been analytically proved and that it is achievable if and only if Hadamard matrix for that n exists. For the cases of n≡1,2,3 (mod 4) sharper bounds have been found. For n=3,5,6,10 we have analytically proved the result using Householder reflections given by Stewart. For the general n two results have been obtained for the problem through Lie groups.