Towards a conjecture concerning minors of Fourier matrices

Andrei Caragea

Session: Frames, Riesz bases, and related topics (Jorge Antezana)

Keywords: Fourier matrix, minors, Chebotarëv’s theorem, exponential basis, permutation

TL;DR: Exploring invertibility of Fourier submatrices.

Abstract:

A result by Chebotarëv states that all minors of
a prime-sized Fourier matrix are non-zero. The authors have
conjectured that for every Fourier matrix of composite size there
exists a permutation of the columns that ensures that all principal
minors of the resulting matrix are non-zero. After correcting
numerical issues, we now know that this conjecture is false for
N = 16. We present some partial results and numerics relating
to this conjecture.

Submission Number: 111