117 lines
3.2 KiB
Text
117 lines
3.2 KiB
Text
/-
|
||
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Leonardo de Moura
|
||
|
||
Matrices
|
||
-/
|
||
import algebra.ring data.fin data.fintype
|
||
open fin nat
|
||
|
||
definition matrix [reducible] (A : Type) (m n : nat) := fin m → fin n → A
|
||
|
||
namespace matrix
|
||
variables {A B C : Type} {m n p : nat}
|
||
|
||
definition val [reducible] (M : matrix A m n) (i : fin m) (j : fin n) : A :=
|
||
M i j
|
||
|
||
namespace ops
|
||
notation M `[` i `, ` j `]` := val M i j
|
||
end ops
|
||
|
||
open ops
|
||
|
||
protected lemma ext {M N : matrix A m n} (h : ∀ i j, M[i,j] = N[i, j]) : M = N :=
|
||
funext (λ i, funext (λ j, h i j))
|
||
|
||
protected lemma has_decidable_eq [h : decidable_eq A] (m n : nat) : decidable_eq (matrix A m n) :=
|
||
_
|
||
|
||
definition to_matrix (f : fin m → fin n → A) : matrix A m n :=
|
||
f
|
||
|
||
definition map (f : A → B) (M : matrix A m n) : matrix B m n :=
|
||
λ i j, f (M[i,j])
|
||
|
||
definition map₂ (f : A → B → C) (M : matrix A m n) (N : matrix B m n) : matrix C m n :=
|
||
λ i j, f (M[i, j]) (N[i,j])
|
||
|
||
definition transpose (M : matrix A m n) : matrix A n m :=
|
||
λ i j, M[j, i]
|
||
|
||
definition symmetric (M : matrix A n n) :=
|
||
transpose M = M
|
||
|
||
section
|
||
variable [r : comm_ring A]
|
||
include r
|
||
|
||
definition identity (n : nat) : matrix A n n :=
|
||
λ i j, if i = j then 1 else 0
|
||
|
||
definition I {n : nat} : matrix A n n :=
|
||
identity n
|
||
|
||
protected definition zero (m n : nat) : matrix A m n :=
|
||
λ i j, 0
|
||
|
||
protected definition add (M : matrix A m n) (N : matrix A m n) : matrix A m n :=
|
||
λ i j, M[i, j] + N[i, j]
|
||
|
||
protected definition sub (M : matrix A m n) (N : matrix A m n) : matrix A m n :=
|
||
λ i j, M[i, j] - N[i, j]
|
||
|
||
protected definition mul (M : matrix A m n) (N : matrix A n p) : matrix A m p :=
|
||
λ i j, fin.foldl has_add.add 0 (λ k : fin n, M[i,k] * N[k,j])
|
||
|
||
definition smul (a : A) (M : matrix A m n) : matrix A m n :=
|
||
λ i j, a * M[i, j]
|
||
|
||
definition matrix_has_zero [reducible] [instance] (m n : nat) : has_zero (matrix A m n) :=
|
||
has_zero.mk (matrix.zero m n)
|
||
|
||
definition matrix_has_one [reducible] [instance] (n : nat) : has_one (matrix A n n) :=
|
||
has_one.mk (identity n)
|
||
|
||
definition matrix_has_add [reducible] [instance] (m n : nat) : has_add (matrix A m n) :=
|
||
has_add.mk matrix.add
|
||
|
||
definition matrix_has_mul [reducible] [instance] (n : nat) : has_mul (matrix A n n) :=
|
||
has_mul.mk matrix.mul
|
||
|
||
infix ` × ` := mul
|
||
infix `⬝` := smul
|
||
|
||
protected lemma add_zero (M : matrix A m n) : M + 0 = M :=
|
||
matrix.ext (λ i j, !add_zero)
|
||
|
||
protected lemma zero_add (M : matrix A m n) : 0 + M = M :=
|
||
matrix.ext (λ i j, !zero_add)
|
||
|
||
protected lemma add.comm (M : matrix A m n) (N : matrix A m n) : M + N = N + M :=
|
||
matrix.ext (λ i j, !add.comm)
|
||
|
||
protected lemma add.assoc (M : matrix A m n) (N : matrix A m n) (P : matrix A m n) : (M + N) + P = M + (N + P) :=
|
||
matrix.ext (λ i j, !add.assoc)
|
||
|
||
definition is_diagonal (M : matrix A n n) :=
|
||
∀ i j, i = j ∨ M[i, j] = 0
|
||
|
||
definition is_zero (M : matrix A m n) :=
|
||
∀ i j, M[i, j] = 0
|
||
|
||
definition is_upper_triangular (M : matrix A n n) :=
|
||
∀ i j : fin n, i > j → M[i, j] = 0
|
||
|
||
definition is_lower_triangular (M : matrix A n n) :=
|
||
∀ i j : fin n, i < j → M[i, j] = 0
|
||
|
||
definition inverse (M : matrix A n n) (N : matrix A n n) :=
|
||
M * N = I ∧ N * M = I
|
||
|
||
definition invertible (M : matrix A n n) :=
|
||
∃ N, inverse M N
|
||
|
||
end
|
||
end matrix
|