On the principal minors of Fourier matrices

TL;DR

The paper investigates when principal minors of the $N$-point Fourier matrix $\mathcal{F}_N$ vanish, revealing a sharp dependence on whether $N$ is square-free. Using complementarity of principal submatrices and index-reduction to small index sets, the authors prove that for $N\ge4$ square-free, all $2\times2$ and $3\times3$ principal minors are nonzero, while for non-square-free $N$ there exist zero minors of every size between $2$ and $N-2$. The proofs combine explicit determinant calculations for $\{0,a\}$ and $\{0,a,b\}$ minors with a trigonometric reformulation that precludes zero determinants in the square-free case, and explicit combinatorial constructions to produce zero minors in the non-square-free case. They also conjecture that, for square-free $N$, all principal minors are nonzero, and discuss connections to Chebotarëv-type results and hierarchical exponential bases relevant to frame theory and Riesz bases.

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.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3: Caragea, Lee, Malikiosis, Pfander
• Proposition 2.1: Proposition 3 in L25
• Lemma 2.2: Proposition 2.13 in CMN24
• proof