Section OneMatrix.
Variable A : 'M[R]_n.
Definition horner_mx := horner_morph (comm_mx_scalar^~ A).
Canonical horner_mx_additive := [additive of horner_mx].
Canonical horner_mx_rmorphism := [rmorphism of horner_mx].
Canonical horner_mx_linear := AddLinear horner_mxZ.
Canonical horner_mx_lrmorphism := [lrmorphism of horner_mx].
Definition powers_mx d := \matrix_(i < d) mxvec (A ^+ i).

Lemma horner_rVpoly m (u : 'rV_m) :
horner_mx (rVpoly u) = vec_mx (u *m powers_mx m).