Spectral Properties of Small Hadamard Matrices

TL;DR

This paper investigates the spectral properties of small Hadamard matrices, proving trace-eigenvalue equivalences for certain sizes and explicitly calculating their spectra, with extensions to larger Fourier permutation matrices.

Contribution

It establishes trace-eigenvalue relationships for 4x4 and 5x5 dephased Hadamard matrices and extends spectral analysis to larger Fourier permutation matrices.

Findings

Trace equality implies identical eigenvalues for 4x4 and 5x5 Hadamard matrices.

Explicit spectra are calculated for these matrices.

Results are extended to larger Fourier permutation matrices.

Abstract

We prove that if $A$ and $B$ are Hadamard matrices which are both of size $4 \times 4$ or $5 \times 5$ and in dephased form, then $tr(A) = tr(B)$ implies that $A$ and $B$ have the same eigenvalues, including multiplicity. We calculate explicitly the spectrum for these matrices. We also extend these results to larger Hadamard matrices which are permutations of the Fourier matrix and calculate their spectral multiplicities.