New classes of matrix decompositions

TL;DR

This paper introduces six new classes of matrix decompositions, demonstrating that any matrix can be expressed as a product of structured matrices, with bounds on the number needed for each class.

Contribution

The paper extends previous work by establishing six novel matrix decomposition classes and proving that any matrix can be factored into these structured matrices with finite products.

Findings

Every n×n matrix is a product of finitely many bidiagonal, skew symmetric, generic, companion, and generalized Vandermonde matrices.

A generic n×n centrosymmetric matrix can be decomposed into a product of symmetric Toeplitz or persymmetric Hankel matrices.

Upper bounds are provided for the number of structured matrices needed in each decomposition.

Abstract

The idea of decomposing a matrix into a product of structured matrices such as triangular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions, extending our work in arXiv:1307.5132. We prove that every $n\times n$ matrix is a product of finitely many bidiagonal, skew symmetric (when n is even), generic, companion matrices and generalized Vandermonde matrices, respectively. We also prove that a generic $n\times n$ centrosymmetric matrix is a product of finitely many symmetric Toeplitz (resp. persymmetric Hankel) matrices. We determine an upper bound of the number of structured matrices needed to decompose a matrix for each case.