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/- | |
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.basic | |
/-! | |
# Orthogonal | |
This file contains definitions and properties concerning orthogonality of rows and columns. | |
## Main results | |
- `matrix.has_orthogonal_rows`: | |
`A.has_orthogonal_rows` means `A` has orthogonal (with respect to `dot_product`) rows. | |
- `matrix.has_orthogonal_cols`: | |
`A.has_orthogonal_cols` means `A` has orthogonal (with respect to `dot_product`) columns. | |
## Tags | |
orthogonal | |
-/ | |
namespace matrix | |
variables {α n m : Type*} | |
variables [has_mul α] [add_comm_monoid α] | |
variables (A : matrix m n α) | |
open_locale matrix | |
/-- `A.has_orthogonal_rows` means matrix `A` has orthogonal rows (with respect to | |
`matrix.dot_product`). -/ | |
def has_orthogonal_rows [fintype n] : Prop := | |
∀ ⦃i₁ i₂⦄, i₁ ≠ i₂ → dot_product (A i₁) (A i₂) = 0 | |
/-- `A.has_orthogonal_rows` means matrix `A` has orthogonal columns (with respect to | |
`matrix.dot_product`). -/ | |
def has_orthogonal_cols [fintype m] : Prop := | |
has_orthogonal_rows Aᵀ | |
/-- `Aᵀ` has orthogonal rows iff `A` has orthogonal columns. -/ | |
@[simp] lemma transpose_has_orthogonal_rows_iff_has_orthogonal_cols [fintype m] : | |
Aᵀ.has_orthogonal_rows ↔ A.has_orthogonal_cols := | |
iff.rfl | |
/-- `Aᵀ` has orthogonal columns iff `A` has orthogonal rows. -/ | |
@[simp] lemma transpose_has_orthogonal_cols_iff_has_orthogonal_rows [fintype n] : | |
Aᵀ.has_orthogonal_cols ↔ A.has_orthogonal_rows := | |
iff.rfl | |
variables {A} | |
lemma has_orthogonal_rows.has_orthogonal_cols | |
[fintype m] (h : Aᵀ.has_orthogonal_rows) : | |
A.has_orthogonal_cols := h | |
lemma has_orthogonal_cols.transpose_has_orthogonal_rows | |
[fintype m] (h : A.has_orthogonal_cols) : | |
Aᵀ.has_orthogonal_rows := h | |
lemma has_orthogonal_cols.has_orthogonal_rows | |
[fintype n] (h : Aᵀ.has_orthogonal_cols) : | |
A.has_orthogonal_rows := h | |
lemma has_orthogonal_rows.transpose_has_orthogonal_cols | |
[fintype n] (h : A.has_orthogonal_rows) : | |
Aᵀ.has_orthogonal_cols := h | |
end matrix | |