Principal non-singularity of Fourier matrices on Zp×ZqZₚ×Zq and Z2k×ZqZ₂ᵏ×Zq

Let F n be the n × n Fourier matrix on the cyclic group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> Zₙ , 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msubsup> </m:math> Zₚ^ ; (ii) the Fourier matrix on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msubsup> <m:mo>×</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:math> 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 reorganized in a unified way. The result will have implications in combining Riesz bases of exponentials.