On the principal minors of Fourier matrices

TL;DR

This paper investigates the conditions under which principal minors of Fourier matrices are nonzero, proving that for square-free dimensions, all 2x2 and 3x3 minors are nonzero, and conjecturing this extends to all minors.

Contribution

It establishes a link between the square-free property of the dimension and the non-vanishing of principal minors of Fourier matrices, and proposes a conjecture for the general case.

Findings

For square-free N ≥ 4, all 2x2 and 3x3 principal minors are nonzero.

If N is not square-free, Fourier matrices have zero principal minors of all sizes.

Numerical experiments suggest all principal minors are nonzero when N is square-free.

Abstract

For the $N$-dimensional Fourier matrix $\mathcal{F}_N$, we prove that if $N\geq 4$ is square-free, then every $2 \times 2$ and $3\times 3$ principal minor of $\mathcal{F}_N$ is nonzero. We also show that if $N\geq 4$ is not square-free, then $\mathcal{F}_N$ has zero principal minors of all sizes. Moreover, based on numerical experiments, we conjecture that if $N$ is square-free, then all principal minors of $\mathcal{F}_N$ are nonzero.