Factoring Matrices into the Product of Circulant and Diagonal Matrices

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.