state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n R
hM : PosSemidef M
z : ℤ
⊢ PosSemidef (M ^ z) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg | protected lemma zpow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (z : ℤ) :
(M ^ z).PosSemidef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.120_0.RRbDg8T8pKv68Qo | protected lemma zpow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (z : ℤ) :
(M ^ z).PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case intro.inl
m : Type u_1
n✝ : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n✝
M : Matrix n✝ n✝ R
hM : PosSemidef M
n : ℕ
⊢ PosSemidef (M ^ ↑n) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa using hM.pow n | protected lemma zpow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (z : ℤ) :
(M ^ z).PosSemidef := by
obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg
· | Mathlib.LinearAlgebra.Matrix.PosDef.120_0.RRbDg8T8pKv68Qo | protected lemma zpow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (z : ℤ) :
(M ^ z).PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case intro.inr
m : Type u_1
n✝ : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n✝
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n✝
M : Matrix n✝ n✝ R
hM : PosSemidef M
n : ℕ
⊢ PosSemidef (M ^ (-↑n)) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa using (hM.pow n).inv | protected lemma zpow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (z : ℤ) :
(M ^ z).PosSemidef := by
obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg
· simpa using hM.pow n
· | Mathlib.LinearAlgebra.Matrix.PosDef.120_0.RRbDg8T8pKv68Qo | protected lemma zpow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (z : ℤ) :
(M ^ z).PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
⊢ PosSemidef hA.sqrt | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | apply PosSemidef.mul_mul_conjTranspose_same | lemma posSemidef_sqrt : PosSemidef hA.sqrt := by
| Mathlib.LinearAlgebra.Matrix.PosDef.160_0.RRbDg8T8pKv68Qo | lemma posSemidef_sqrt : PosSemidef hA.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
case hA
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
⊢ PosSemidef (diagonal (IsROrC.ofReal ∘ Real.sqrt ∘ IsHermitian.eigenvalu... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine posSemidef_diagonal_iff.mpr fun i ↦ ?_ | lemma posSemidef_sqrt : PosSemidef hA.sqrt := by
apply PosSemidef.mul_mul_conjTranspose_same
| Mathlib.LinearAlgebra.Matrix.PosDef.160_0.RRbDg8T8pKv68Qo | lemma posSemidef_sqrt : PosSemidef hA.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
case hA
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
i : n
⊢ 0 ≤ (IsROrC.ofReal ∘ Real.sqrt ∘ IsHermitian.eigenvalues (_ : IsH... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [Function.comp_apply, IsROrC.nonneg_iff] | lemma posSemidef_sqrt : PosSemidef hA.sqrt := by
apply PosSemidef.mul_mul_conjTranspose_same
refine posSemidef_diagonal_iff.mpr fun i ↦ ?_
| Mathlib.LinearAlgebra.Matrix.PosDef.160_0.RRbDg8T8pKv68Qo | lemma posSemidef_sqrt : PosSemidef hA.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
case hA
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
i : n
⊢ 0 ≤ IsROrC.re ↑((Real.sqrt ∘ IsHermitian.eigenvalues (_ : IsHermi... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | constructor | lemma posSemidef_sqrt : PosSemidef hA.sqrt := by
apply PosSemidef.mul_mul_conjTranspose_same
refine posSemidef_diagonal_iff.mpr fun i ↦ ?_
rw [Function.comp_apply, IsROrC.nonneg_iff]
| Mathlib.LinearAlgebra.Matrix.PosDef.160_0.RRbDg8T8pKv68Qo | lemma posSemidef_sqrt : PosSemidef hA.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
case hA.left
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
i : n
⊢ 0 ≤ IsROrC.re ↑((Real.sqrt ∘ IsHermitian.eigenvalues (_ : Is... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only [IsROrC.ofReal_re] | lemma posSemidef_sqrt : PosSemidef hA.sqrt := by
apply PosSemidef.mul_mul_conjTranspose_same
refine posSemidef_diagonal_iff.mpr fun i ↦ ?_
rw [Function.comp_apply, IsROrC.nonneg_iff]
constructor
· | Mathlib.LinearAlgebra.Matrix.PosDef.160_0.RRbDg8T8pKv68Qo | lemma posSemidef_sqrt : PosSemidef hA.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
case hA.left
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
i : n
⊢ 0 ≤ (Real.sqrt ∘ IsHermitian.eigenvalues (_ : IsHermitian A)... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact Real.sqrt_nonneg _ | lemma posSemidef_sqrt : PosSemidef hA.sqrt := by
apply PosSemidef.mul_mul_conjTranspose_same
refine posSemidef_diagonal_iff.mpr fun i ↦ ?_
rw [Function.comp_apply, IsROrC.nonneg_iff]
constructor
· simp only [IsROrC.ofReal_re]
| Mathlib.LinearAlgebra.Matrix.PosDef.160_0.RRbDg8T8pKv68Qo | lemma posSemidef_sqrt : PosSemidef hA.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
case hA.right
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
i : n
⊢ IsROrC.im ↑((Real.sqrt ∘ IsHermitian.eigenvalues (_ : IsHer... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only [IsROrC.ofReal_im] | lemma posSemidef_sqrt : PosSemidef hA.sqrt := by
apply PosSemidef.mul_mul_conjTranspose_same
refine posSemidef_diagonal_iff.mpr fun i ↦ ?_
rw [Function.comp_apply, IsROrC.nonneg_iff]
constructor
· simp only [IsROrC.ofReal_re]
exact Real.sqrt_nonneg _
· | Mathlib.LinearAlgebra.Matrix.PosDef.160_0.RRbDg8T8pKv68Qo | lemma posSemidef_sqrt : PosSemidef hA.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
⊢ hA.sqrt ^ 2 = A | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | let C := hA.1.eigenvectorMatrix | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
| Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
⊢ hA.sqrt ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜) | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
| Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
E : Matrix... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
| Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
E : Matrix... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [Matrix.PosSemidef.sqrt, pow_two] | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
E : Matrix... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | change (C * E * Cᴴ) * (C * E * Cᴴ) = A | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
| Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
E : Matrix... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [← mul_assoc] using this | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
| Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | have : Cᴴ * C = 1 | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
| Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [Matrix.IsHermitian.conjTranspose_eigenvectorMatrix, mul_eq_one_comm] | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact hA.1.eigenvectorMatrix_mul_inv | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [this, mul_one] | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | have : E * E = diagonal ((↑) ∘ hA.1.eigenvalues) | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [diagonal_mul_diagonal] | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine congr_arg _ (funext fun v ↦ ?_) | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp [← pow_two, ← IsROrC.ofReal_pow, Real.sq_sqrt (hA.eigenvalues_nonneg v)] | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [this] | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case this
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHermitian A)
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | convert hA.1.spectral_theorem'.symm | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
case h.e'_2.h.e'_6
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
C : Matrix n n 𝕜 := IsHermitian.eigenvectorMatrix (_ : IsHerm... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | apply Matrix.IsHermitian.conjTranspose_eigenvectorMatrix | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A := by
let C := hA.1.eigenvectorMatrix
let E := diagonal ((↑) ∘ Real.sqrt ∘ hA.1.eigenvalues : n → 𝕜)
suffices : C * (E * (Cᴴ * C) * E) * Cᴴ = A
· rw [Matrix.PosSemidef.sqrt, pow_two]
change (C * E * Cᴴ) * (C * E * Cᴴ) = A
simpa only [← mul_assoc] using this
hav... | Mathlib.LinearAlgebra.Matrix.PosDef.169_0.RRbDg8T8pKv68Qo | @[simp]
lemma sq_sqrt : hA.sqrt ^ 2 = A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
⊢ hA.sqrt * hA.sqrt = A | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [← pow_two, sq_sqrt] | @[simp]
lemma sqrt_mul_self : hA.sqrt * hA.sqrt = A := by | Mathlib.LinearAlgebra.Matrix.PosDef.189_0.RRbDg8T8pKv68Qo | @[simp]
lemma sqrt_mul_self : hA.sqrt * hA.sqrt = A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
⊢ A = B | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | by_contra h_ne | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
⊢ False | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | let ⟨v, t, ht, hv, hv'⟩ := (hA.1.sub hB.1).exists_eigenvector_of_ne_zero (sub_ne_zero.mpr h_ne) | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | have h_sum : 0 = t * (star v ⬝ᵥ mulVec A v + star v ⬝ᵥ mulVec B v) | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case h_sum
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | calc
0 = star v ⬝ᵥ mulVec (A ^ 2 - B ^ 2) v := by rw [hAB, sub_self, zero_mulVec, dotProduct_zero]
_ = star v ⬝ᵥ mulVec A (mulVec (A - B) v) + star v ⬝ᵥ mulVec (A - B) (mulVec B v) := by
rw [mulVec_mulVec, mulVec_mulVec, ← dotProduct_add, ← add_mulVec, mul_sub, sub_mul,
add_sub, sub_add_c... | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hAB, sub_self, zero_mulVec, dotProduct_zero] | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [mulVec_mulVec, mulVec_mulVec, ← dotProduct_add, ← add_mulVec, mul_sub, sub_mul,
add_sub, sub_add_cancel, pow_two, pow_two] | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hv', mulVec_smul, dotProduct_smul, IsROrC.real_smul_eq_coe_mul,
dotProduct_mulVec _ (A - B), hA.1.sub hB.1] | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp_rw [← star_mulVec, hv', mul_add, ← IsROrC.real_smul_eq_coe_mul, ← smul_dotProduct] | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | congr 2 with i | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case e_a.e_v.h
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only [Pi.star_apply, Pi.smul_apply, IsROrC.real_smul_eq_coe_mul, star_mul',
IsROrC.star_def, IsROrC.conj_ofReal] | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | replace h_sum : star v ⬝ᵥ mulVec A v + star v ⬝ᵥ mulVec B v = 0 | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case h_sum
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [eq_comm, ← mul_zero (t : 𝕜)] at h_sum | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case h_sum
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact mul_left_cancel₀ (IsROrC.ofReal_ne_zero.mpr ht) h_sum | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | have h_van : star v ⬝ᵥ mulVec A v = 0 ∧ star v ⬝ᵥ mulVec B v = 0 | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case h_van
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine ⟨le_antisymm ?_ (hA.2 v), le_antisymm ?_ (hB.2 v)⟩ | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case h_van.refine_1
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [add_comm, add_eq_zero_iff_eq_neg] at h_sum | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case h_van.refine_1
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [h_sum, neg_nonneg] using hB.2 v | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case h_van.refine_2
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [add_eq_zero_iff_eq_neg.mp h_sum, neg_nonneg] using hA.2 v | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | have aux : star v ⬝ᵥ mulVec (A - B) v = 0 | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
case aux
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [sub_mulVec, dotProduct_sub, h_van.1, h_van.2, sub_zero] | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hv', dotProduct_smul, IsROrC.real_smul_eq_coe_mul, ← mul_zero ↑t] at aux | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B ^ 2
h_ne : ¬A = B
v : n → 𝕜
... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact hv <| Matrix.dotProduct_star_self_eq_zero.mp <| mul_left_cancel₀
(IsROrC.ofReal_ne_zero.mpr ht) aux | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B := by
/- This is deceptively hard, much more difficult than the positive *definite* case. We follow a
clever proof due to Koeber and Schäfer. The idea is that if `A ≠ B`, then `A - B` has a nonzero
real eigenvalue, with eig... | Mathlib.LinearAlgebra.Matrix.PosDef.192_0.RRbDg8T8pKv68Qo | lemma eq_of_sq_eq_sq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B ^ 2) : A = B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
B : Matrix n n 𝕜
hB : PosSemidef B
hAB : A ^ 2 = B
⊢ A = hB.sqrt | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | subst B | lemma eq_sqrt_of_sq_eq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B) : A = hB.sqrt := by
| Mathlib.LinearAlgebra.Matrix.PosDef.232_0.RRbDg8T8pKv68Qo | lemma eq_sqrt_of_sq_eq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B) : A = hB.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
hB : PosSemidef (A ^ 2)
⊢ A = hB.sqrt | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hA.sqrt_sq] | lemma eq_sqrt_of_sq_eq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B) : A = hB.sqrt := by
subst B
| Mathlib.LinearAlgebra.Matrix.PosDef.232_0.RRbDg8T8pKv68Qo | lemma eq_sqrt_of_sq_eq {B : Matrix n n 𝕜} (hB : PosSemidef B) (hAB : A ^ 2 = B) : A = hB.sqrt | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
e : m ≃ n
h : PosSemidef (submatrix M ⇑e ⇑e)
⊢ PosSemidef M | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa using h.submatrix e.symm | @[simp]
theorem posSemidef_submatrix_equiv {M : Matrix n n R} (e : m ≃ n) :
(M.submatrix e e).PosSemidef ↔ M.PosSemidef :=
⟨fun h => by | Mathlib.LinearAlgebra.Matrix.PosDef.240_0.RRbDg8T8pKv68Qo | @[simp]
theorem posSemidef_submatrix_equiv {M : Matrix n n R} (e : m ≃ n) :
(M.submatrix e e).PosSemidef ↔ M.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
A : Matrix m n R
⊢ PosSemidef (Aᴴ * A) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine ⟨isHermitian_transpose_mul_self _, fun x => ?_⟩ | /-- The conjugate transpose of a matrix mulitplied by the matrix is positive semidefinite -/
theorem posSemidef_conjTranspose_mul_self (A : Matrix m n R) : PosSemidef (Aᴴ * A) := by
| Mathlib.LinearAlgebra.Matrix.PosDef.246_0.RRbDg8T8pKv68Qo | /-- The conjugate transpose of a matrix mulitplied by the matrix is positive semidefinite -/
theorem posSemidef_conjTranspose_mul_self (A : Matrix m n R) : PosSemidef (Aᴴ * A) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
A : Matrix m n R
x : n → R
⊢ 0 ≤ star x ⬝ᵥ mulVec (Aᴴ * A) x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [← mulVec_mulVec, dotProduct_mulVec, vecMul_conjTranspose, star_star] | /-- The conjugate transpose of a matrix mulitplied by the matrix is positive semidefinite -/
theorem posSemidef_conjTranspose_mul_self (A : Matrix m n R) : PosSemidef (Aᴴ * A) := by
refine ⟨isHermitian_transpose_mul_self _, fun x => ?_⟩
| Mathlib.LinearAlgebra.Matrix.PosDef.246_0.RRbDg8T8pKv68Qo | /-- The conjugate transpose of a matrix mulitplied by the matrix is positive semidefinite -/
theorem posSemidef_conjTranspose_mul_self (A : Matrix m n R) : PosSemidef (Aᴴ * A) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
A : Matrix m n R
x : n → R
⊢ 0 ≤ star (mulVec A x) ⬝ᵥ mulVec A x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact Finset.sum_nonneg fun i _ => star_mul_self_nonneg _ | /-- The conjugate transpose of a matrix mulitplied by the matrix is positive semidefinite -/
theorem posSemidef_conjTranspose_mul_self (A : Matrix m n R) : PosSemidef (Aᴴ * A) := by
refine ⟨isHermitian_transpose_mul_self _, fun x => ?_⟩
rw [← mulVec_mulVec, dotProduct_mulVec, vecMul_conjTranspose, star_star]
| Mathlib.LinearAlgebra.Matrix.PosDef.246_0.RRbDg8T8pKv68Qo | /-- The conjugate transpose of a matrix mulitplied by the matrix is positive semidefinite -/
theorem posSemidef_conjTranspose_mul_self (A : Matrix m n R) : PosSemidef (Aᴴ * A) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
A : Matrix m n R
⊢ PosSemidef (A * Aᴴ) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [conjTranspose_conjTranspose] using posSemidef_conjTranspose_mul_self Aᴴ | /-- A matrix multiplied by its conjugate transpose is positive semidefinite -/
theorem posSemidef_self_mul_conjTranspose (A : Matrix m n R) : PosSemidef (A * Aᴴ) :=
by | Mathlib.LinearAlgebra.Matrix.PosDef.252_0.RRbDg8T8pKv68Qo | /-- A matrix multiplied by its conjugate transpose is positive semidefinite -/
theorem posSemidef_self_mul_conjTranspose (A : Matrix m n R) : PosSemidef (A * Aᴴ) | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
⊢ PosSemidef A ↔ ∃ B, A = Bᴴ * B | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine ⟨fun hA ↦ ⟨hA.sqrt, ?_⟩, fun ⟨B, hB⟩ ↦ (hB ▸ posSemidef_conjTranspose_mul_self B)⟩ | /-- A matrix is positive semidefinite if and only if it has the form `Bᴴ * B` for some `B`. -/
lemma posSemidef_iff_eq_transpose_mul_self [DecidableEq n] {A : Matrix n n 𝕜} :
PosSemidef A ↔ ∃ (B : Matrix n n 𝕜), A = Bᴴ * B := by
| Mathlib.LinearAlgebra.Matrix.PosDef.264_0.RRbDg8T8pKv68Qo | /-- A matrix is positive semidefinite if and only if it has the form `Bᴴ * B` for some `B`. -/
lemma posSemidef_iff_eq_transpose_mul_self [DecidableEq n] {A : Matrix n n 𝕜} :
PosSemidef A ↔ ∃ (B : Matrix n n 𝕜), A = Bᴴ * B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
⊢ A = hA.sqrtᴴ * hA.sqrt | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp_rw [← PosSemidef.sq_sqrt hA, pow_two] | /-- A matrix is positive semidefinite if and only if it has the form `Bᴴ * B` for some `B`. -/
lemma posSemidef_iff_eq_transpose_mul_self [DecidableEq n] {A : Matrix n n 𝕜} :
PosSemidef A ↔ ∃ (B : Matrix n n 𝕜), A = Bᴴ * B := by
refine ⟨fun hA ↦ ⟨hA.sqrt, ?_⟩, fun ⟨B, hB⟩ ↦ (hB ▸ posSemidef_conjTranspose_mul_se... | Mathlib.LinearAlgebra.Matrix.PosDef.264_0.RRbDg8T8pKv68Qo | /-- A matrix is positive semidefinite if and only if it has the form `Bᴴ * B` for some `B`. -/
lemma posSemidef_iff_eq_transpose_mul_self [DecidableEq n] {A : Matrix n n 𝕜} :
PosSemidef A ↔ ∃ (B : Matrix n n 𝕜), A = Bᴴ * B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
⊢ hA.sqrt * hA.sqrt = hA.sqrtᴴ * hA.sqrt | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hA.posSemidef_sqrt.1] | /-- A matrix is positive semidefinite if and only if it has the form `Bᴴ * B` for some `B`. -/
lemma posSemidef_iff_eq_transpose_mul_self [DecidableEq n] {A : Matrix n n 𝕜} :
PosSemidef A ↔ ∃ (B : Matrix n n 𝕜), A = Bᴴ * B := by
refine ⟨fun hA ↦ ⟨hA.sqrt, ?_⟩, fun ⟨B, hB⟩ ↦ (hB ▸ posSemidef_conjTranspose_mul_se... | Mathlib.LinearAlgebra.Matrix.PosDef.264_0.RRbDg8T8pKv68Qo | /-- A matrix is positive semidefinite if and only if it has the form `Bᴴ * B` for some `B`. -/
lemma posSemidef_iff_eq_transpose_mul_self [DecidableEq n] {A : Matrix n n 𝕜} :
PosSemidef A ↔ ∃ (B : Matrix n n 𝕜), A = Bᴴ * B | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : IsHermitian A
h : ∀ (i : n), 0 ≤ eigenvalues hA i
⊢ PosSemidef A | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp_rw [hA.conjTranspose_eigenvectorMatrix.symm ▸ hA.spectral_theorem'] | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A := by
| Mathlib.LinearAlgebra.Matrix.PosDef.271_0.RRbDg8T8pKv68Qo | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : IsHermitian A
h : ∀ (i : n), 0 ≤ eigenvalues hA i
⊢ PosSemidef (eigenvectorMatrix hA * diagona... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine (posSemidef_diagonal_iff.mpr fun i ↦ ?_).mul_mul_conjTranspose_same _ | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A := by
simp_rw [hA.conjTranspose_eigenvectorMatrix.symm ▸ hA.spectral_theorem']
| Mathlib.LinearAlgebra.Matrix.PosDef.271_0.RRbDg8T8pKv68Qo | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : IsHermitian A
h : ∀ (i : n), 0 ≤ eigenvalues hA i
i : n
⊢ 0 ≤ (IsROrC.ofReal ∘ eigenvalues hA)... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [IsROrC.le_iff_re_im] | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A := by
simp_rw [hA.conjTranspose_eigenvectorMatrix.symm ▸ hA.spectral_theorem']
refine (posSemidef_diagonal_iff.mpr fun i ↦ ?_).mul_mul_conjTranspose_same... | Mathlib.LinearAlgebra.Matrix.PosDef.271_0.RRbDg8T8pKv68Qo | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : IsHermitian A
h : ∀ (i : n), 0 ≤ eigenvalues hA i
i : n
⊢ IsROrC.re 0 ≤ IsROrC.re ((IsROrC.ofR... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa using h i | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A := by
simp_rw [hA.conjTranspose_eigenvectorMatrix.symm ▸ hA.spectral_theorem']
refine (posSemidef_diagonal_iff.mpr fun i ↦ ?_).mul_mul_conjTranspose_same... | Mathlib.LinearAlgebra.Matrix.PosDef.271_0.RRbDg8T8pKv68Qo | lemma IsHermitian.posSemidef_of_eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : IsHermitian A) (h : ∀ i : n, 0 ≤ hA.eigenvalues i) : PosSemidef A | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
x : n → 𝕜
⊢ star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | constructor | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 := by
| Mathlib.LinearAlgebra.Matrix.PosDef.278_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
case mp
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
x : n → 𝕜
⊢ star x ⬝ᵥ mulVec A x = 0 → mulVec A x = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | obtain ⟨B, rfl⟩ := posSemidef_iff_eq_transpose_mul_self.mp hA | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 := by
constructor
· | Mathlib.LinearAlgebra.Matrix.PosDef.278_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
case mp.intro
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
x : n → 𝕜
B : Matrix n n 𝕜
hA : PosSemidef (Bᴴ * B)
⊢ star x ⬝ᵥ mulVec (Bᴴ * B) x = 0 → mulVec (Bᴴ * ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [← Matrix.mulVec_mulVec, dotProduct_mulVec,
vecMul_conjTranspose, star_star, dotProduct_star_self_eq_zero] | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 := by
constructor
· obtain ⟨B, rfl⟩ := posSemidef_iff_eq_transpose_mul_self... | Mathlib.LinearAlgebra.Matrix.PosDef.278_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
case mp.intro
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
x : n → 𝕜
B : Matrix n n 𝕜
hA : PosSemidef (Bᴴ * B)
⊢ mulVec B x = 0 → mulVec Bᴴ (mulVec B x) = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | intro h0 | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 := by
constructor
· obtain ⟨B, rfl⟩ := posSemidef_iff_eq_transpose_mul_self... | Mathlib.LinearAlgebra.Matrix.PosDef.278_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
case mp.intro
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
x : n → 𝕜
B : Matrix n n 𝕜
hA : PosSemidef (Bᴴ * B)
h0 : mulVec B x = 0
⊢ mulVec Bᴴ (mulVec B x) = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [h0, mulVec_zero] | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 := by
constructor
· obtain ⟨B, rfl⟩ := posSemidef_iff_eq_transpose_mul_self... | Mathlib.LinearAlgebra.Matrix.PosDef.278_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
case mpr
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
x : n → 𝕜
⊢ mulVec A x = 0 → star x ⬝ᵥ mulVec A x = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | intro h0 | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 := by
constructor
· obtain ⟨B, rfl⟩ := posSemidef_iff_eq_transpose_mul_self... | Mathlib.LinearAlgebra.Matrix.PosDef.278_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
case mpr
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
x : n → 𝕜
h0 : mulVec A x = 0
⊢ star x ⬝ᵥ mulVec A x = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [h0, dotProduct_zero] | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 := by
constructor
· obtain ⟨B, rfl⟩ := posSemidef_iff_eq_transpose_mul_self... | Mathlib.LinearAlgebra.Matrix.PosDef.278_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0`. -/
theorem PosSemidef.dotProduct_mulVec_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
star x ⬝ᵥ mulVec A x = 0 ↔ mulVec A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosSemidef A
x : n → 𝕜
⊢ ((toLinearMap₂' A) (star x)) x = 0 ↔ (toLin' A) x = 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [toLinearMap₂'_apply', toLin'_apply] using hA.dotProduct_mulVec_zero_iff x | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0` (linear maps version). -/
theorem PosSemidef.toLinearMap₂'_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
Matrix.toLinearMap₂' A (star x) x = 0 ↔ Matrix.toLin' A x = 0 := by
| Mathlib.LinearAlgebra.Matrix.PosDef.291_0.RRbDg8T8pKv68Qo | /-- For `A` positive semidefinite, we have `x⋆ A x = 0` iff `A x = 0` (linear maps version). -/
theorem PosSemidef.toLinearMap₂'_zero_iff [DecidableEq n]
{A : Matrix n n 𝕜} (hA : PosSemidef A) (x : n → 𝕜) :
Matrix.toLinearMap₂' A (star x) x = 0 ↔ Matrix.toLin' A x = 0 | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
⊢ PosSemidef M | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine' ⟨hM.1, _⟩ | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.317_0.RRbDg8T8pKv68Qo | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
⊢ ∀ (x : n → R), 0 ≤ star x ⬝ᵥ mulVec M x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | intro x | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef := by
refine' ⟨hM.1, _⟩
| Mathlib.LinearAlgebra.Matrix.PosDef.317_0.RRbDg8T8pKv68Qo | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
x : n → R
⊢ 0 ≤ star x ⬝ᵥ mulVec M x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | by_cases hx : x = 0 | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef := by
refine' ⟨hM.1, _⟩
intro x
| Mathlib.LinearAlgebra.Matrix.PosDef.317_0.RRbDg8T8pKv68Qo | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case pos
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
x : n → R
hx : x = 0
⊢ 0 ≤ star x ⬝ᵥ mulVec M x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only [hx, zero_dotProduct, star_zero, IsROrC.zero_re'] | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef := by
refine' ⟨hM.1, _⟩
intro x
by_cases hx : x = 0
· | Mathlib.LinearAlgebra.Matrix.PosDef.317_0.RRbDg8T8pKv68Qo | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case pos
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
x : n → R
hx : x = 0
⊢ 0 ≤ 0 | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact le_rfl | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef := by
refine' ⟨hM.1, _⟩
intro x
by_cases hx : x = 0
· simp only [hx, zero_dotProduct, star_zero, IsROrC.zero_re']
| Mathlib.LinearAlgebra.Matrix.PosDef.317_0.RRbDg8T8pKv68Qo | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
case neg
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
x : n → R
hx : ¬x = 0
⊢ 0 ≤ star x ⬝ᵥ mulVec M x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact le_of_lt (hM.2 x hx) | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef := by
refine' ⟨hM.1, _⟩
intro x
by_cases hx : x = 0
· simp only [hx, zero_dotProduct, star_zero, IsROrC.zero_re']
exact le_rfl
· | Mathlib.LinearAlgebra.Matrix.PosDef.317_0.RRbDg8T8pKv68Qo | theorem posSemidef {M : Matrix n n R} (hM : M.PosDef) : M.PosSemidef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
⊢ PosDef Mᵀ | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine ⟨IsHermitian.transpose hM.1, fun x hx => ?_⟩ | theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.326_0.RRbDg8T8pKv68Qo | theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
x : n → R
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ mulVec Mᵀ x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | convert hM.2 (star x) (star_ne_zero.2 hx) using 1 | theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef := by
refine ⟨IsHermitian.transpose hM.1, fun x hx => ?_⟩
| Mathlib.LinearAlgebra.Matrix.PosDef.326_0.RRbDg8T8pKv68Qo | theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
case h.e'_4
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarOrderedRing R
inst✝ : IsROrC 𝕜
M : Matrix n n R
hM : PosDef M
x : n → R
hx : x ≠ 0
⊢ star x ⬝ᵥ mulVec Mᵀ x = star (star x) ⬝ᵥ mulVec M (star x) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [mulVec_transpose, Matrix.dotProduct_mulVec, star_star, dotProduct_comm] | theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef := by
refine ⟨IsHermitian.transpose hM.1, fun x hx => ?_⟩
convert hM.2 (star x) (star_ne_zero.2 hx) using 1
| Mathlib.LinearAlgebra.Matrix.PosDef.326_0.RRbDg8T8pKv68Qo | theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : IsSymm M
hMq : QuadraticForm.PosDef (toQuadraticForm' M)
⊢ PosDef M | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine' ⟨hM, fun x hx => _⟩ | theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
(hMq : M.toQuadraticForm'.PosDef) : M.PosDef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.332_0.RRbDg8T8pKv68Qo | theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
(hMq : M.toQuadraticForm'.PosDef) : M.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : IsSymm M
hMq : QuadraticForm.PosDef (toQuadraticForm' M)
x : n → ℝ
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ m... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only [toQuadraticForm', QuadraticForm.PosDef, BilinForm.toQuadraticForm_apply,
Matrix.toBilin'_apply'] at hMq | theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
(hMq : M.toQuadraticForm'.PosDef) : M.PosDef := by
refine' ⟨hM, fun x hx => _⟩
| Mathlib.LinearAlgebra.Matrix.PosDef.332_0.RRbDg8T8pKv68Qo | theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
(hMq : M.toQuadraticForm'.PosDef) : M.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : IsSymm M
x : n → ℝ
hx : x ≠ 0
hMq : ∀ (x : n → ℝ), x ≠ 0 → 0 < x ⬝ᵥ mulVec M x
⊢ 0 < star x ⬝ᵥ ... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | apply hMq x hx | theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
(hMq : M.toQuadraticForm'.PosDef) : M.PosDef := by
refine' ⟨hM, fun x hx => _⟩
simp only [toQuadraticForm', QuadraticForm.PosDef, BilinForm.toQuadraticForm_apply,
Matrix.toBilin'_apply'] at hMq
| Mathlib.LinearAlgebra.Matrix.PosDef.332_0.RRbDg8T8pKv68Qo | theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
(hMq : M.toQuadraticForm'.PosDef) : M.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
⊢ QuadraticForm.PosDef (Matrix.toQuadraticForm' M) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | intro x hx | theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
M.toQuadraticForm'.PosDef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.340_0.RRbDg8T8pKv68Qo | theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
M.toQuadraticForm'.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
x : n → ℝ
hx : x ≠ 0
⊢ 0 < (Matrix.toQuadraticForm' M) x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simp only [Matrix.toQuadraticForm', BilinForm.toQuadraticForm_apply, Matrix.toBilin'_apply'] | theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
M.toQuadraticForm'.PosDef := by
intro x hx
| Mathlib.LinearAlgebra.Matrix.PosDef.340_0.RRbDg8T8pKv68Qo | theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
M.toQuadraticForm'.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
x : n → ℝ
hx : x ≠ 0
⊢ 0 < x ⬝ᵥ mulVec M x | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | apply hM.2 x hx | theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
M.toQuadraticForm'.PosDef := by
intro x hx
simp only [Matrix.toQuadraticForm', BilinForm.toQuadraticForm_apply, Matrix.toBilin'_apply']
| Mathlib.LinearAlgebra.Matrix.PosDef.340_0.RRbDg8T8pKv68Qo | theorem toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :
M.toQuadraticForm'.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosDef A
i : n
⊢ 0 < IsHermitian.eigenvalues (_ : IsHermitian A) i | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hA.1.eigenvalues_eq, hA.1.transpose_eigenvectorMatrix_apply] | /-- The eigenvalues of a positive definite matrix are positive -/
lemma eigenvalues_pos [DecidableEq n] {A : Matrix n n 𝕜}
(hA : Matrix.PosDef A) (i : n) : 0 < hA.1.eigenvalues i := by
| Mathlib.LinearAlgebra.Matrix.PosDef.347_0.RRbDg8T8pKv68Qo | /-- The eigenvalues of a positive definite matrix are positive -/
lemma eigenvalues_pos [DecidableEq n] {A : Matrix n n 𝕜}
(hA : Matrix.PosDef A) (i : n) : 0 < hA.1.eigenvalues i | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : PosDef A
i : n
⊢ 0 <
IsROrC.re
(star ((IsHermitian.eigenvectorBasis (_ : IsHermitian... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact hA.re_dotProduct_pos <| hA.1.eigenvectorBasis.orthonormal.ne_zero i | /-- The eigenvalues of a positive definite matrix are positive -/
lemma eigenvalues_pos [DecidableEq n] {A : Matrix n n 𝕜}
(hA : Matrix.PosDef A) (i : n) : 0 < hA.1.eigenvalues i := by
rw [hA.1.eigenvalues_eq, hA.1.transpose_eigenvectorMatrix_apply]
| Mathlib.LinearAlgebra.Matrix.PosDef.347_0.RRbDg8T8pKv68Qo | /-- The eigenvalues of a positive definite matrix are positive -/
lemma eigenvalues_pos [DecidableEq n] {A : Matrix n n 𝕜}
(hA : Matrix.PosDef A) (i : n) : 0 < hA.1.eigenvalues i | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
⊢ 0 < det M | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hM.isHermitian.det_eq_prod_eigenvalues] | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M := by
| Mathlib.LinearAlgebra.Matrix.PosDef.353_0.RRbDg8T8pKv68Qo | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M | Mathlib_LinearAlgebra_Matrix_PosDef |
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
⊢ 0 < Finset.prod Finset.univ fun i => ↑(IsHermitian.eigenvalues (_ : IsHermitian M) i... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | apply Finset.prod_pos | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M := by
rw [hM.isHermitian.det_eq_prod_eigenvalues]
| Mathlib.LinearAlgebra.Matrix.PosDef.353_0.RRbDg8T8pKv68Qo | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M | Mathlib_LinearAlgebra_Matrix_PosDef |
case h0
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
⊢ ∀ i ∈ Finset.univ, 0 < ↑(IsHermitian.eigenvalues (_ : IsHermitian M) i) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | intro i _ | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M := by
rw [hM.isHermitian.det_eq_prod_eigenvalues]
apply Finset.prod_pos
| Mathlib.LinearAlgebra.Matrix.PosDef.353_0.RRbDg8T8pKv68Qo | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M | Mathlib_LinearAlgebra_Matrix_PosDef |
case h0
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
i : n
a✝ : i ∈ Finset.univ
⊢ 0 < ↑(IsHermitian.eigenvalues (_ : IsHermitian M)... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [hM.isHermitian.eigenvalues_eq] | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M := by
rw [hM.isHermitian.det_eq_prod_eigenvalues]
apply Finset.prod_pos
intro i _
| Mathlib.LinearAlgebra.Matrix.PosDef.353_0.RRbDg8T8pKv68Qo | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M | Mathlib_LinearAlgebra_Matrix_PosDef |
case h0
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
i : n
a✝ : i ∈ Finset.univ
⊢ 0 <
↑(IsROrC.re
(star ((IsHermitian.e... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | refine hM.2 _ fun h => ?_ | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M := by
rw [hM.isHermitian.det_eq_prod_eigenvalues]
apply Finset.prod_pos
intro i _
rw [hM.isHermitian.eigenvalues_eq]
| Mathlib.LinearAlgebra.Matrix.PosDef.353_0.RRbDg8T8pKv68Qo | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M | Mathlib_LinearAlgebra_Matrix_PosDef |
case h0
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
i : n
a✝ : i ∈ Finset.univ
h : (IsHermitian.eigenvectorMatrix (_ : IsHermitian... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | have h_det : hM.isHermitian.eigenvectorMatrixᵀ.det = 0 :=
Matrix.det_eq_zero_of_row_eq_zero i fun j => congr_fun h j | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M := by
rw [hM.isHermitian.det_eq_prod_eigenvalues]
apply Finset.prod_pos
intro i _
rw [hM.isHermitian.eigenvalues_eq]
refine hM.2 _ fun h => ?_
| Mathlib.LinearAlgebra.Matrix.PosDef.353_0.RRbDg8T8pKv68Qo | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M | Mathlib_LinearAlgebra_Matrix_PosDef |
case h0
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarOrderedRing R
inst✝¹ : IsROrC 𝕜
inst✝ : DecidableEq n
M : Matrix n n ℝ
hM : PosDef M
i : n
a✝ : i ∈ Finset.univ
h : (IsHermitian.eigenvectorMatrix (_ : IsHermitian... | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | simpa only [h_det, not_isUnit_zero] using
isUnit_det_of_invertible hM.isHermitian.eigenvectorMatrixᵀ | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M := by
rw [hM.isHermitian.det_eq_prod_eigenvalues]
apply Finset.prod_pos
intro i _
rw [hM.isHermitian.eigenvalues_eq]
refine hM.2 _ fun h => ?_
have h_det : hM.isHermitian.eigenvectorMatrixᵀ.det = 0 :=
Matrix.det_eq_zero_of_ro... | Mathlib.LinearAlgebra.Matrix.PosDef.353_0.RRbDg8T8pKv68Qo | theorem det_pos [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : 0 < det M | Mathlib_LinearAlgebra_Matrix_PosDef |
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
Q : QuadraticForm ℝ (n → ℝ)
hQ : Matrix.PosDef (toMatrix' Q)
⊢ PosDef Q | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [← toQuadraticForm_associated ℝ Q,
← BilinForm.toMatrix'.left_inv ((associatedHom (R := ℝ) ℝ) Q)] | theorem posDef_of_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)}
(hQ : Q.toMatrix'.PosDef) : Q.PosDef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.373_0.RRbDg8T8pKv68Qo | theorem posDef_of_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)}
(hQ : Q.toMatrix'.PosDef) : Q.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
Q : QuadraticForm ℝ (n → ℝ)
hQ : Matrix.PosDef (toMatrix' Q)
⊢ PosDef
(BilinForm.toQuadraticForm
(LinearEquiv.invFun BilinForm.toMatrix' (AddHom.toFun (↑BilinForm.toMatrix').toAddHom ((associatedHom ℝ) Q)))) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact hQ.toQuadraticForm' | theorem posDef_of_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)}
(hQ : Q.toMatrix'.PosDef) : Q.PosDef := by
rw [← toQuadraticForm_associated ℝ Q,
← BilinForm.toMatrix'.left_inv ((associatedHom (R := ℝ) ℝ) Q)]
| Mathlib.LinearAlgebra.Matrix.PosDef.373_0.RRbDg8T8pKv68Qo | theorem posDef_of_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)}
(hQ : Q.toMatrix'.PosDef) : Q.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
Q : QuadraticForm ℝ (n → ℝ)
hQ : PosDef Q
⊢ Matrix.PosDef (toMatrix' Q) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | rw [← toQuadraticForm_associated ℝ Q, ←
BilinForm.toMatrix'.left_inv ((associatedHom (R := ℝ) ℝ) Q)] at hQ | theorem posDef_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)} (hQ : Q.PosDef) :
Q.toMatrix'.PosDef := by
| Mathlib.LinearAlgebra.Matrix.PosDef.380_0.RRbDg8T8pKv68Qo | theorem posDef_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)} (hQ : Q.PosDef) :
Q.toMatrix'.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
n : Type u_1
inst✝¹ : Fintype n
inst✝ : DecidableEq n
Q : QuadraticForm ℝ (n → ℝ)
hQ :
PosDef
(BilinForm.toQuadraticForm
(LinearEquiv.invFun BilinForm.toMatrix' (AddHom.toFun (↑BilinForm.toMatrix').toAddHom ((associatedHom ℝ) Q))))
⊢ Matrix.PosDef (toMatrix' Q) | /-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | exact .of_toQuadraticForm' (isSymm_toMatrix' Q) hQ | theorem posDef_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)} (hQ : Q.PosDef) :
Q.toMatrix'.PosDef := by
rw [← toQuadraticForm_associated ℝ Q, ←
BilinForm.toMatrix'.left_inv ((associatedHom (R := ℝ) ℝ) Q)] at hQ
| Mathlib.LinearAlgebra.Matrix.PosDef.380_0.RRbDg8T8pKv68Qo | theorem posDef_toMatrix' [DecidableEq n] {Q : QuadraticForm ℝ (n → ℝ)} (hQ : Q.PosDef) :
Q.toMatrix'.PosDef | Mathlib_LinearAlgebra_Matrix_PosDef |
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