Principal non-singularity of Fourier matrices on Zp×ZqZ₏Zₐ and Z2k×ZqZ₂^kZₐ

Abstract Let F n be the n × n Fourier matrix on the cyclic group Z n Z₍, a renowned theorem of Chebotarëv asserts that all minors in F n for prime n are non-zero. In this short note it is shown that (i) all principal minors in the Kronecker product F p ⊗ F q are non-vanishing (principal non-singularity) for distinct odd primes p, q if q is large enough and generates the multiplicative group Z p * Z₏^{} ; (ii) the Fourier matrix on Z 2 k × Z q Z₂^kZₐ 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 reorganized in a unified way. The result will have implications in combining Riesz bases of exponentials.