Eigenvalues and eigenvectors of complex Hadamard matrices

TL;DR

This paper explores the eigenvalues and eigenvectors of complex Hadamard matrices, providing new bounds and properties that could aid in their complete classification, especially for the 6x6 case.

Contribution

It establishes bounds on eigenvalue multiplicities of 6x6 complex Hadamard matrices and extends these results to arbitrary dimensions, advancing classification efforts.

Findings

Any dephased n×n CHM has two constant eigenvalues ±√n.

Maximum number of identical eigenvalues in 6×6 CHMs with dephased form is determined.

No 6×6 CHM has four identical eigenvalues.

Abstract

Characterizing the $6\times 6$ complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. In this paper, we investigate the eigenvalues and eigenvectors of CHMs. We show that any $n\times n$ CHM with dephased form has two constant eigenvalues $\pm\sqrt{n}$ and has two constant eigenvectors. We obtain the maximum numbers of identical eigenvalues of $6\times 6$ CHMs with dephased form and we extend this result to arbitrary dimension. We also show that there is no $6\times 6$ CHM with four identical eigenvalues. We conjecture that the eigenvalues and eigenvectors of $6\times 6$ CHMs will lead to the complete classification of $6\times 6$ CHMs.