A note on companion matrices

We show that the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices, n−1 of them being the identity matrix in which a 2×2 identity submatrix in two consecutive rows (and columns) is replaced by an appropriate 2×2 matrix, the remaining being the identity matrix with the last entry replaced by possibly different entry. By a certain similarity transformation, we obtain a simple new companion matrix in a pentadiagonal form. Some generalizations are also possible.