109 lines
2.7 KiB
Text
109 lines
2.7 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 algebra 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
|
|||
|
|
|
|||
|
|
definition zero (m n : nat) : matrix A m n :=
|
|||
|
|
λ i j, 0
|
|||
|
|
|
|||
|
|
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]
|
|||
|
|
|
|||
|
|
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]
|
|||
|
|
|
|||
|
|
definition smul (a : A) (M : matrix A m n) : matrix A m n :=
|
|||
|
|
λ i j, a * M[i, j]
|
|||
|
|
|
|||
|
|
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])
|
|||
|
|
|
|||
|
|
infix + := add
|
|||
|
|
infix - := sub
|
|||
|
|
infix * := mul
|
|||
|
|
infix * := smul
|
|||
|
|
notation 0 := zero _ _
|
|||
|
|
|
|||
|
|
lemma add_zero (M : matrix A m n) : M + 0 = M :=
|
|||
|
|
matrix.ext (λ i j, !algebra.add_zero)
|
|||
|
|
|
|||
|
|
lemma zero_add (M : matrix A m n) : 0 + M = M :=
|
|||
|
|
matrix.ext (λ i j, !algebra.zero_add)
|
|||
|
|
|
|||
|
|
lemma add.comm (M : matrix A m n) (N : matrix A m n) : M + N = N + M :=
|
|||
|
|
matrix.ext (λ i j, !algebra.add.comm)
|
|||
|
|
|
|||
|
|
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, !algebra.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, i > j → M[i, j] = 0
|
|||
|
|
|
|||
|
|
definition is_lower_triangular (M : matrix A n n) :=
|
|||
|
|
∀ i j, 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
|