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/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import linear_algebra.matrix.spectrum
import linear_algebra.quadratic_form.basic
/-! # Positive Definite Matrices
This file defines positive definite matrices and connects this notion to positive definiteness of
quadratic forms.
## Main definition
* `matrix.pos_def` : a matrix `M : matrix n n R` is positive definite if it is hermitian
and `xᴴMx` is greater than zero for all nonzero `x`.
-/
namespace matrix
variables {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
open_locale matrix
/-- A matrix `M : matrix n n R` is positive definite if it is hermitian
and `xᴴMx` is greater than zero for all nonzero `x`. -/
def pos_def (M : matrix n n 𝕜) :=
M.is_hermitian ∧ ∀ x : n → 𝕜, x ≠ 0 → 0 < is_R_or_C.re (dot_product (star x) (M.mul_vec x))
lemma pos_def.is_hermitian {M : matrix n n 𝕜} (hM : M.pos_def) : M.is_hermitian := hM.1
lemma pos_def_of_to_quadratic_form' [decidable_eq n] {M : matrix n n ℝ}
(hM : M.is_symm) (hMq : M.to_quadratic_form'.pos_def) :
M.pos_def :=
begin
refine ⟨hM, λ x hx, _⟩,
simp only [to_quadratic_form', quadratic_form.pos_def, bilin_form.to_quadratic_form_apply,
matrix.to_bilin'_apply'] at hMq,
apply hMq x hx,
end
lemma pos_def_to_quadratic_form' [decidable_eq n] {M : matrix n n ℝ} (hM : M.pos_def) :
M.to_quadratic_form'.pos_def :=
begin
intros x hx,
simp only [to_quadratic_form', bilin_form.to_quadratic_form_apply, matrix.to_bilin'_apply'],
apply hM.2 x hx,
end
end matrix
namespace quadratic_form
variables {n : Type*} [fintype n]
lemma pos_def_of_to_matrix'
[decidable_eq n] {Q : quadratic_form ℝ (n → ℝ)} (hQ : Q.to_matrix'.pos_def) :
Q.pos_def :=
begin
rw [←to_quadratic_form_associated ℝ Q,
←bilin_form.to_matrix'.left_inv ((associated_hom _) Q)],
apply matrix.pos_def_to_quadratic_form' hQ
end
lemma pos_def_to_matrix' [decidable_eq n] {Q : quadratic_form ℝ (n → ℝ)} (hQ : Q.pos_def) :
Q.to_matrix'.pos_def :=
begin
rw [←to_quadratic_form_associated ℝ Q,
←bilin_form.to_matrix'.left_inv ((associated_hom _) Q)] at hQ,
apply matrix.pos_def_of_to_quadratic_form' (is_symm_to_matrix' Q) hQ,
end
end quadratic_form
namespace matrix
variables {𝕜 : Type*} [is_R_or_C 𝕜] {n : Type*} [fintype n]
/-- A positive definite matrix `M` induces an inner product `⟪x, y⟫ = xᴴMy`. -/
noncomputable def inner_product_space.of_matrix
{M : matrix n n 𝕜} (hM : M.pos_def) : inner_product_space 𝕜 (n → 𝕜) :=
inner_product_space.of_core
{ inner := λ x y, dot_product (star x) (M.mul_vec y),
conj_sym := λ x y, by
rw [star_dot_product, star_ring_end_apply, star_star, star_mul_vec,
dot_product_mul_vec, hM.is_hermitian.eq],
nonneg_re := λ x,
begin
by_cases h : x = 0,
{ simp [h] },
{ exact le_of_lt (hM.2 x h) }
end,
definite := λ x hx,
begin
by_contra' h,
simpa [hx, lt_self_iff_false] using hM.2 x h,
end,
add_left := by simp only [star_add, add_dot_product, eq_self_iff_true, forall_const],
smul_left := λ x y r, by rw [← smul_eq_mul, ←smul_dot_product, star_ring_end_apply, ← star_smul] }
end matrix
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