Canonical forms for polynomial and quadratic differential operators

We define the canonical representative for the equivalence class consisting of all polynomial- and quadratic differential operators that take the same values on a given subspace of C∞(Rn,Rw)C∞(Rn,Rw). We also examine the relation of this definition with that given in a special case in [J.C. Willems, H.L. Trentelman, On quadratic differential forms, SIAM J. Control Optim. 36 (1998) 1703–1749].