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.
Weiqi Zhou (Fri,) studied this question.