Informing the structure of complex Hadamard matrix spaces using a flow

TL;DR

This paper introduces a novel dynamical systems approach using gradient flows and the Center Manifold Theorem to analyze the local structure and construct families of complex Hadamard matrices.

Contribution

It offers a new interpretation of the defect as a center subspace dimension and applies this to study the local structure of Hadamard matrix spaces.

Findings

New interpretation of defect as center subspace dimension

Application of dynamical systems theory to Hadamard matrices

Construction of affine families of Hadamard matrices

Abstract

The defect of a complex Hadamard matrix $H$ is an upper bound for the dimension of a continuous Hadamard orbit stemming from $H$. We provide a new interpretation of the defect as the dimension of the center subspace of a gradient flow and apply the Center Manifold Theorem of dynamical systems theory to study local structure in spaces of complex Hadamard matrices. Through examples, we provide several applications of our methodology including the construction of affine families of Hadamard matrices.