# Factoring Matrices into the Product of Circulant and Diagonal Matrices

Marko Huhtanen*, Allan Perämäki

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

7 Citations (Scopus)

## 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.

Original language English 1018-1033 16 Journal of Fourier Analysis and Applications 21 5 https://doi.org/10.1007/s00041-015-9395-0 Published - 26 Mar 2015 A1 Journal article-refereed

## Keywords

• Circulant matrix
• Diagonal matrix
• Matrix factoring
• Multiplicative Fourier compression
• Polynomial factoring
• Sparsity structure