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Rank-deficient submatrices of Kronecker products of Fourier matrices. (English) Zbl 1124.42008
Summary: We provide a set of maximal rank-deficient submatrices of a Kronecker product of two matrices $A\otimes B$, and in particular the Kronecker product of Fourier matrices $F={F}_{{n}_{1}}\otimes \cdots \otimes {F}_{{n}_{k}}$. We show how in the latter case, maximal rank-deficient submatrices can be constructed as tilings of rank-one blocks. Several such tilings may be associated to any subgroup of the Abelian group ${ℤ}_{{n}_{1}}×\cdots ×{ℤ}_{{n}_{k}}$ that corresponds to the matrix $F$. The maximal rank-deficient submatrices of $F$ are also related to an uncertainty principle for Fourier transforms over finite Abelian groups, for which we can then obtain stronger versions.
##### MSC:
 42A99 Fourier analysis in one variable 15A03 Vector spaces, linear dependence, rank 15A69 Multilinear algebra, tensor products
##### Keywords:
uncertainty principle