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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 AB, and in particular the Kronecker product of Fourier matrices F=F n 1 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 n 1 ×× 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:
42A99Fourier analysis in one variable
15A03Vector spaces, linear dependence, rank
15A69Multilinear algebra, tensor products