New classes of matrix decompositions

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.