Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 4,116 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import data.matrix.block
/-!
# Symmetric matrices
This file contains the definition and basic results about symmetric matrices.
## Main definition
* `matrix.is_symm `: a matrix `A : matrix n n α` is "symmetric" if `Aᵀ = A`.
## Tags
symm, symmetric, matrix
-/
variables {α β n m R : Type*}
namespace matrix
open_locale matrix
/-- A matrix `A : matrix n n α` is "symmetric" if `Aᵀ = A`. -/
def is_symm (A : matrix n n α) : Prop := Aᵀ = A
lemma is_symm.eq {A : matrix n n α} (h : A.is_symm) : Aᵀ = A := h
/-- A version of `matrix.ext_iff` that unfolds the `matrix.transpose`. -/
lemma is_symm.ext_iff {A : matrix n n α} : A.is_symm ↔ ∀ i j, A j i = A i j :=
matrix.ext_iff.symm
/-- A version of `matrix.ext` that unfolds the `matrix.transpose`. -/
@[ext]
lemma is_symm.ext {A : matrix n n α} : (∀ i j, A j i = A i j) → A.is_symm :=
matrix.ext
lemma is_symm.apply {A : matrix n n α} (h : A.is_symm) (i j : n) : A j i = A i j :=
is_symm.ext_iff.1 h i j
lemma is_symm_mul_transpose_self [fintype n] [comm_semiring α] (A : matrix n n α) :
(A ⬝ Aᵀ).is_symm :=
transpose_mul _ _
lemma is_symm_transpose_mul_self [fintype n] [comm_semiring α] (A : matrix n n α) :
(Aᵀ ⬝ A).is_symm :=
transpose_mul _ _
lemma is_symm_add_transpose_self [add_comm_semigroup α] (A : matrix n n α) :
(A + Aᵀ).is_symm :=
add_comm _ _
lemma is_symm_transpose_add_self [add_comm_semigroup α] (A : matrix n n α) :
(Aᵀ + A).is_symm :=
add_comm _ _
@[simp] lemma is_symm_zero [has_zero α] :
(0 : matrix n n α).is_symm :=
transpose_zero
@[simp] lemma is_symm_one [decidable_eq n] [has_zero α] [has_one α] :
(1 : matrix n n α).is_symm :=
transpose_one
@[simp] lemma is_symm.map {A : matrix n n α} (h : A.is_symm) (f : α → β) :
(A.map f).is_symm :=
transpose_map.symm.trans (h.symm ▸ rfl)
@[simp] lemma is_symm.transpose {A : matrix n n α} (h : A.is_symm) :
Aᵀ.is_symm :=
congr_arg _ h
@[simp] lemma is_symm.conj_transpose [has_star α] {A : matrix n n α} (h : A.is_symm) :
Aᴴ.is_symm :=
h.transpose.map _
@[simp] lemma is_symm.neg [has_neg α] {A : matrix n n α} (h : A.is_symm) :
(-A).is_symm :=
(transpose_neg _).trans (congr_arg _ h)
@[simp] lemma is_symm.add {A B : matrix n n α} [has_add α] (hA : A.is_symm) (hB : B.is_symm) :
(A + B).is_symm :=
(transpose_add _ _).trans (hA.symm ▸ hB.symm ▸ rfl)
@[simp] lemma is_symm.sub {A B : matrix n n α} [has_sub α] (hA : A.is_symm) (hB : B.is_symm) :
(A - B).is_symm :=
(transpose_sub _ _).trans (hA.symm ▸ hB.symm ▸ rfl)
@[simp] lemma is_symm.smul [has_smul R α] {A : matrix n n α} (h : A.is_symm) (k : R) :
(k • A).is_symm :=
(transpose_smul _ _).trans (congr_arg _ h)
@[simp] lemma is_symm.minor {A : matrix n n α} (h : A.is_symm) (f : m → n) :
(A.minor f f).is_symm :=
(transpose_minor _ _ _).trans (h.symm ▸ rfl)
/-- The diagonal matrix `diagonal v` is symmetric. -/
@[simp] lemma is_symm_diagonal [decidable_eq n] [has_zero α] (v : n → α) :
(diagonal v).is_symm :=
diagonal_transpose _
/-- A block matrix `A.from_blocks B C D` is symmetric,
if `A` and `D` are symmetric and `Bᵀ = C`. -/
lemma is_symm.from_blocks
{A : matrix m m α} {B : matrix m n α} {C : matrix n m α} {D : matrix n n α}
(hA : A.is_symm) (hBC : Bᵀ = C) (hD : D.is_symm) :
(A.from_blocks B C D).is_symm :=
begin
have hCB : Cᵀ = B, {rw ← hBC, simp},
unfold matrix.is_symm,
rw from_blocks_transpose,
congr;
assumption
end
/-- This is the `iff` version of `matrix.is_symm.from_blocks`. -/
lemma is_symm_from_blocks_iff
{A : matrix m m α} {B : matrix m n α} {C : matrix n m α} {D : matrix n n α} :
(A.from_blocks B C D).is_symm ↔ A.is_symm ∧ Bᵀ = C ∧ Cᵀ = B ∧ D.is_symm :=
⟨λ h, ⟨congr_arg to_blocks₁₁ h, congr_arg to_blocks₂₁ h,
congr_arg to_blocks₁₂ h, congr_arg to_blocks₂₂ h⟩,
λ ⟨hA, hBC, hCB, hD⟩, is_symm.from_blocks hA hBC hD⟩
end matrix
|