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Alizter merged 4 commits into from
Jun 3, 2024
Merged

theory of diagonal matrices #1977

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c7689a4
theory of diagonal matrices
Alizter Jun 1, 2024
40e5962
speed up matrix_diag_ring
Alizter Jun 3, 2024
b312fba
reivew suggestions
Alizter Jun 3, 2024
fde5a49
generalise path_entry_vector
Alizter Jun 3, 2024
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19 changes: 7 additions & 12 deletions theories/Algebra/Rings/Matrix.v
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Original file line number Diff line number Diff line change
Expand Up @@ -310,7 +310,7 @@ Defined.

(** ** Diagonal matrices *)

(** A diagonal matrix is a matrix with zeros everywhere except on the diagonal. It's entries are given by a vector. *)
(** A diagonal matrix is a matrix with zeros everywhere except on the diagonal. Its entries are given by a vector. *)
Definition matrix_diag {R : Ring@{i}} {n : nat} (v : Vector R n)
: Matrix R n n.
Proof.
Expand All @@ -319,20 +319,15 @@ Proof.
exact (kronecker_delta i j * Vector.entry v i).
Defined.

(** Addition of matrices is the same as addition of the corresponding vectors. *)
(** Addition of diagonal matrices is the same as addition of the corresponding vectors. *)
Definition matrix_diag_plus {R : Ring@{i}} {n : nat} (v w : Vector R n)
: matrix_plus (matrix_diag v) (matrix_diag w) = matrix_diag (vector_plus v w).
Proof.
symmetry.
snrapply path_matrix.
intros i j Hi Hj.
rewrite 2 entry_Build_Matrix, 4 entry_Build_Vector.
cbv beta zeta; destruct (dec (i = j)) as [p | np].
- destruct p.
rewrite kronecker_delta_refl.
rewrite <- rng_dist_l.
by rewrite entry_Build_Vector.
- rewrite (kronecker_delta_neq np).
by rewrite 3 rng_mult_zero_l, rng_plus_zero_l.
rewrite 2 entry_Build_Matrix, 5 entry_Build_Vector.
nrapply rng_dist_l.
Defined.

(** Matrix multiplication of diagonal matrices is the same as multiplying the corresponding vectors pointwise. *)
Expand All @@ -347,7 +342,7 @@ Proof.
{ intros k Hk.
rewrite 2 entry_Build_Matrix.
rewrite rng_mult_assoc.
rewrite <- (rng_mult_assoc (kronecker_delta _ _ )).
rewrite <- (rng_mult_assoc (kronecker_delta _ _)).
rewrite kronecker_delta_comm.
rewrite <- 2 rng_mult_assoc.
reflexivity. }
Expand All @@ -365,7 +360,7 @@ Proof.
rewrite kronecker_delta_symm.
unfold kronecker_delta.
destruct (dec (i = j)) as [p|np].
1: destruct p; f_ap; f_ap; apply path_ishprop.
1: destruct p; f_ap; apply path_entry_vector.
by rewrite !rng_mult_zero_l.
Defined.

Expand Down
7 changes: 7 additions & 0 deletions theories/Algebra/Rings/Vector.v
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Expand Up @@ -61,6 +61,13 @@ Proof.
1, 2: apply nth'_nth'.
Defined.

Definition path_entry_vector {A : Type} {n : nat} (v : Vector A n)
(i : nat) (Hi Hi' : (i < n)%nat)
: @entry _ _ v i Hi = @entry _ _ v i Hi'.
Proof.
apply nth'_nth'.
Defined.

(** ** Operations *)

Definition vector_map {A B : Type} {n} (f : A -> B)
Expand Down