Principal Non-singularity of Fourier Matrices on Zₚ × Zq and Z₂ᵏ × Zq

Let Fₙ be the n× n Fourier matrix (on cyclic groups Zₙ), a reknowned theorem of Chebotarëv asserts that all minors in Fₙ for prime n are non-zero. In this short note it is shown that (i) all principal minors in the Kronecker product Fₚ⊗ Fq are non-vanishing (principal non-singularity) for distinct odd primes $p,q$ if q is large enough and generates the multiplicative group Zₚ^*; (ii) the Fourier matrix on Z₂ᵏ × Zq is principally non-singular upon permutation (in particular, for $k=1$ the identity permutation suffices) for odd prime q and $k=1,2,3$. The proof is just an exposition of existing techniques re-organized in a unified way. The result will have implications in combining Riesz bases of exponentials.