Section StructuralLinear.
Canonical diag_mx_additive n := Additive (@diag_mx_is_linear n).
Canonical diag_mx_linear n := Linear (@diag_mx_is_linear n).
(* Matrix multiplication using bigops. *)
Fact mulmx_key : unit. Proof. by []. Qed.
Definition mulmx {m n p} (A : 'M_(m, n)) (B : 'M_(n, p)) : 'M[R]_(m, p) :=
\matrix[mulmx_key]_(i, k) \sum_j (A i j * B j k).
Local Notation "A *m B" := (mulmx A B) : ring_scope.
(* Right scaling associativity requires a commutative ring *)
(* Partial identity matrix (used in rank decomposition). *)
Fact pid_mx_key : unit. Proof. by []. Qed.
Definition pid_mx {m n} r : 'M[R]_(m, n) :=
\matrix[pid_mx_key]_(i, j) ((i == j :> nat) && (i < r))%:R.

Lemma mul_col_mx m1 m2 n p (Au : 'M_(m1, n)) (Ad : 'M_(m2, n)) (B : 'M_(n, p)) :
col_mx Au Ad *m B = col_mx (Au *m B) (Ad *m B).