  A decomposition of a higher order linear differential operator with
polynomial coefficients into a direct sum of two factor operators is obtained.
This leads to a lower echelon matrix representation for operators of the above
mentioned type arising from a finite sequence of elementary operations. As a
consequence, a complete basis of canonical polynomials is generated via a
modified Ortiz' recurrence formula. The approach presented in this paper
enables us to curry out the full and self-operative recursive implementation of
the spectral-tau method. These results can be applied to a variety of operator
equations associated with infinite matrices in lower row echelon form, having
implications for a broad family of discrete and continuous numerical methods,
interpreted as special realizations of the tau method under specific
perturbation terms.
