# 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.

Research output: Contribution to journal › Article › Scientific › peer-review

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*Journal of Fourier Analysis and Applications*, vol. 21, no. 5, pp. 1018-1033. https://doi.org/10.1007/s00041-015-9395-0

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*Journal of Fourier Analysis and Applications*,

*21*(5), 1018-1033. https://doi.org/10.1007/s00041-015-9395-0

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TY - JOUR

T1 - Factoring Matrices into the Product of Circulant and Diagonal Matrices

AU - Huhtanen, Marko

AU - Perämäki, Allan

PY - 2015/3/26

Y1 - 2015/3/26

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.

KW - Circulant matrix

KW - Diagonal matrix

KW - Matrix factoring

KW - Multiplicative Fourier compression

KW - Polynomial factoring

KW - Sparsity structure

UR - http://www.scopus.com/inward/record.url?scp=84941024966&partnerID=8YFLogxK

U2 - 10.1007/s00041-015-9395-0

DO - 10.1007/s00041-015-9395-0

M3 - Article

VL - 21

SP - 1018

EP - 1033

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

SN - 1069-5869

IS - 5

ER -

ID: 10194797