Factoring Matrices into the Product of Circulant and Diagonal Matrices

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Factoring Matrices into the Product of Circulant and Diagonal Matrices. / Huhtanen, Marko; Perämäki, Allan.

In: Journal of Fourier Analysis and Applications, Vol. 21, No. 5, 26.03.2015, p. 1018-1033.

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Huhtanen, Marko ; Perämäki, Allan. / Factoring Matrices into the Product of Circulant and Diagonal Matrices. In: Journal of Fourier Analysis and Applications. 2015 ; Vol. 21, No. 5. pp. 1018-1033.

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title = "Factoring Matrices into the Product of Circulant and Diagonal Matrices",
abstract = "A generic matrix $$A\in \,\mathbb {C}^{n \times n}$$A∈Cn×n is shown to be the product of circulant and diagonal matrices with the number of factors being $$2n-1$$2n-1 at most. The demonstration is constructive, relying on first factoring matrix subspaces equivalent to polynomials in a permutation matrix over diagonal matrices into linear factors. For the linear factors, the sum of two scaled permutations is factored into the product of a circulant matrix and two diagonal matrices. Extending the monomial group, both low degree and sparse polynomials in a permutation matrix over diagonal matrices, together with their permutation equivalences, constitute a fundamental sparse matrix structure. Matrix analysis gets largely done polynomially, in terms of permutations only.",
keywords = "Circulant matrix, Diagonal matrix, Matrix factoring, Multiplicative Fourier compression, Polynomial factoring, Sparsity structure",
author = "Marko Huhtanen and Allan Per{\"a}m{\"a}ki",
year = "2015",
month = "3",
day = "26",
doi = "10.1007/s00041-015-9395-0",
language = "English",
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pages = "1018--1033",
journal = "Journal of Fourier Analysis and Applications",
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number = "5",

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N2 - A generic matrix $$A\in \,\mathbb {C}^{n \times n}$$A∈Cn×n is shown to be the product of circulant and diagonal matrices with the number of factors being $$2n-1$$2n-1 at most. The demonstration is constructive, relying on first factoring matrix subspaces equivalent to polynomials in a permutation matrix over diagonal matrices into linear factors. For the linear factors, the sum of two scaled permutations is factored into the product of a circulant matrix and two diagonal matrices. Extending the monomial group, both low degree and sparse polynomials in a permutation matrix over diagonal matrices, together with their permutation equivalences, constitute a fundamental sparse matrix structure. Matrix analysis gets largely done polynomially, in terms of permutations only.

AB - A generic matrix $$A\in \,\mathbb {C}^{n \times n}$$A∈Cn×n is shown to be the product of circulant and diagonal matrices with the number of factors being $$2n-1$$2n-1 at most. The demonstration is constructive, relying on first factoring matrix subspaces equivalent to polynomials in a permutation matrix over diagonal matrices into linear factors. For the linear factors, the sum of two scaled permutations is factored into the product of a circulant matrix and two diagonal matrices. Extending the monomial group, both low degree and sparse polynomials in a permutation matrix over diagonal matrices, together with their permutation equivalences, constitute a fundamental sparse matrix structure. Matrix analysis gets largely done polynomially, in terms of permutations only.

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KW - Diagonal matrix

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KW - Multiplicative Fourier compression

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