Abstract
A generic matrix \(A\in \,\mathbb {C}^{n \times n}\) is shown to be the product of circulant and diagonal matrices with the number of factors being \(2n-1\) 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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Notes
In particular, any unitary matrix \(A\in \,\mathbb {C}^{n \times n}\) can be interpreted as being a diffractive optical system. See [15] how products of discrete Fourier transforms and diagonal matrices model diffractive optical elements.
The cyclic shift of size \(n\)-by-\(n\) has ones below the main diagonal and at the position \((1,n)\).
This is a more operator theoretic formulation admitting an extension to infinite dimensions [12].
This approach certainly works in the generic case of \(D\) having differing diagonal entries in the absolute value. In this paper we do not consider the numerical recovering of whether \(A\) belongs to (2.2) in general.
A standard basis matrix of \(\,\mathbb {C}^{n \times n}\) has exactly one entry equaling one while its other entries are zeros.
Block diagonal matrices are used, e.g., in preconditioning. Thereby the sum of two block PD matrices is certainly of interest by providing a more flexible preconditioning structure.
The aim of preprocessing depends, to some degree, on whether one uses iterative methods or sparse direct methods; see [3, p. 438].
When the dimension is two, one essentially deals with a generalized eigenvalue problem. For solving generalized eigenvalue problems there are reliable numerical methods.
It would be tempting to call such a \((0,1)\)-matrix a polynomial digraph. It has, however, another meaning [2, p.157].
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Communicated by Yang Wang.
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Huhtanen, M., Perämäki, A. Factoring Matrices into the Product of Circulant and Diagonal Matrices. J Fourier Anal Appl 21, 1018–1033 (2015). https://doi.org/10.1007/s00041-015-9395-0
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DOI: https://doi.org/10.1007/s00041-015-9395-0
Keywords
- Circulant matrix
- Diagonal matrix
- Sparsity structure
- Matrix factoring
- Polynomial factoring
- Multiplicative Fourier compression