Entropy and Hadamard Matrices

TL;DR

This paper explores the entropy of orthogonal matrices, revealing Hadamard matrices as entropy-maximizing solutions and proposing their interpretation as Morse functions with notable maxima and saddle points.

Contribution

It introduces a novel entropy-based perspective on Hadamard matrices and analyzes their critical points as Morse functions on the group manifold.

Findings

Hadamard matrices maximize entropy in their class

Maxima for 3 and 5 dimensions are explicitly presented

Identifies Morse function properties of entropy on matrix groups

Abstract

The entropy of an orthogonal matrix is defined. It provides a new interpretation of Hadamard matrices as those that saturate the bound for entropy.It appears to be a useful Morse function on the group manifold. It has sharp maxima and other saddle points. The matrices corresponding to the maxima for 3 and 5 dimensions are presented. They are integer matrices (upto a rescaling).