  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.
