fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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noncomputable basisFun : Basis η R (η → R) :=
Basis.ofEquivFun (LinearEquiv.refl _ _)
@[simp] | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basisFun | The basis on `η → R` where the `i`th basis vector is `Function.update 0 i 1`. |
basisFun_apply [DecidableEq η] (i) :
basisFun R η i = Pi.single i 1 := by
simp only [basisFun, Basis.coe_ofEquivFun, LinearEquiv.refl_symm, LinearEquiv.refl_apply]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basisFun_apply | null |
basisFun_repr (x : η → R) (i : η) : (Pi.basisFun R η).repr x i = x i := by simp [basisFun]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basisFun_repr | null |
basisFun_equivFun : (Pi.basisFun R η).equivFun = LinearEquiv.refl _ _ :=
Basis.equivFun_ofEquivFun _ | theorem | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | basisFun_equivFun | null |
AlgHom.eq_piEvalAlgHom {k G : Type*} [CommSemiring k] [NoZeroDivisors k] [Nontrivial k]
[Finite G] (φ : (G → k) →ₐ[k] k) : ∃ (s : G), φ = Pi.evalAlgHom _ _ s := by
have h1 := map_one φ
classical
have := Fintype.ofFinite G
simp only [← Finset.univ_sum_single (1 : G → k), Pi.one_apply, map_sum] at h1
obtain ⟨s, hs⟩ : ∃ (s : G), φ (Pi.single s 1) ≠ 0 := by
by_contra
simp_all
have h2 : ∀ t ≠ s, φ (Pi.single t 1) = 0 := by
refine fun _ _ ↦ (eq_zero_or_eq_zero_of_mul_eq_zero ?_).resolve_left hs
rw [← map_mul]
convert map_zero φ
ext u
by_cases u = s <;> simp_all
have h3 : φ (Pi.single s 1) = 1 := by
rwa [Fintype.sum_eq_single s h2] at h1
use s
refine AlgHom.toLinearMap_injective ((Pi.basisFun k G).ext fun t ↦ ?_)
by_cases t = s <;> simp_all
@[deprecated (since := "2025-04-15")] alias eval_of_algHom := AlgHom.eq_piEvalAlgHom | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | AlgHom.eq_piEvalAlgHom | Let `k` be an integral domain and `G` an arbitrary finite set.
Then any algebra morphism `φ : (G → k) →ₐ[k] k` is an evaluation map. |
noncomputable piEquiv : (ι → M) ≃ₗ[R] ((ι → R) →ₗ[R] M) := Basis.constr (Pi.basisFun R ι) R | def | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | piEquiv | The natural linear equivalence: `Mⁱ ≃ Hom(Rⁱ, M)` for an `R`-module `M`. |
piEquiv_apply_apply (ι R M : Type*) [Fintype ι] [CommSemiring R]
[AddCommMonoid M] [Module R M] (v : ι → M) (w : ι → R) :
piEquiv ι R M v w = ∑ i, w i • v i := by
simp only [piEquiv, Basis.constr_apply_fintype, Basis.equivFun_apply]
congr
@[simp] lemma range_piEquiv (v : ι → M) :
LinearMap.range (piEquiv ι R M v) = span R (range v) :=
Basis.constr_range _ _
@[simp] lemma surjective_piEquiv_apply_iff (v : ι → M) :
Surjective (piEquiv ι R M v) ↔ span R (range v) = ⊤ := by
rw [← LinearMap.range_eq_top, range_piEquiv] | lemma | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | piEquiv_apply_apply | null |
_root_.Module.Free.pi (M : ι → Type*) [Finite ι] [∀ i : ι, AddCommMonoid (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] : Module.Free R (∀ i, M i) :=
let ⟨_⟩ := nonempty_fintype ι
.of_basis <| Pi.basis fun i => Module.Free.chooseBasis R (M i)
variable (ι) in | instance | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | _root_.Module.Free.pi | The product of finitely many free modules is free. |
_root_.Module.Free.function [Finite ι] [Module.Free R M] : Module.Free R (ι → M) :=
Free.pi _ _ | instance | LinearAlgebra | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.LinearAlgebra.Finsupp.VectorSpace",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearIndependent.Lemmas"
] | Mathlib/LinearAlgebra/StdBasis.lean | _root_.Module.Free.function | The product of finitely many free modules is free (non-dependent version to help with typeclass
search). |
J : Matrix (l ⊕ l) (l ⊕ l) R :=
Matrix.fromBlocks 0 (-1) 1 0
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | J | The matrix defining the canonical skew-symmetric bilinear form. |
J_transpose : (J l R)ᵀ = -J l R := by
rw [J, fromBlocks_transpose, ← neg_one_smul R (fromBlocks _ _ _ _ : Matrix (l ⊕ l) (l ⊕ l) R),
fromBlocks_smul, Matrix.transpose_zero, Matrix.transpose_one, transpose_neg]
simp [fromBlocks]
variable [Fintype l] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | J_transpose | null |
J_squared : J l R * J l R = -1 := by
rw [J, fromBlocks_multiply]
simp only [Matrix.zero_mul, Matrix.neg_mul, zero_add, neg_zero, Matrix.one_mul, add_zero]
rw [← neg_zero, ← Matrix.fromBlocks_neg, ← fromBlocks_one] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | J_squared | null |
J_inv : (J l R)⁻¹ = -J l R := by
refine Matrix.inv_eq_right_inv ?_
rw [Matrix.mul_neg, J_squared]
exact neg_neg 1 | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | J_inv | null |
J_det_mul_J_det : det (J l R) * det (J l R) = 1 := by
rw [← det_mul, J_squared, ← one_smul R (-1 : Matrix _ _ R), smul_neg, ← neg_smul, det_smul,
Fintype.card_sum, det_one, mul_one]
apply Even.neg_one_pow
exact Even.add_self _ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | J_det_mul_J_det | null |
isUnit_det_J : IsUnit (det (J l R)) :=
isUnit_iff_exists_inv.mpr ⟨det (J l R), J_det_mul_J_det _ _⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | isUnit_det_J | null |
symplecticGroup : Submonoid (Matrix (l ⊕ l) (l ⊕ l) R) where
carrier := { A | A * J l R * Aᵀ = J l R }
mul_mem' {a b} ha hb := by
simp only [Set.mem_setOf_eq, transpose_mul] at *
rw [← Matrix.mul_assoc, a.mul_assoc, a.mul_assoc, hb]
exact ha
one_mem' := by simp | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | symplecticGroup | The group of symplectic matrices over a ring `R`. |
mem_iff {A : Matrix (l ⊕ l) (l ⊕ l) R} :
A ∈ symplecticGroup l R ↔ A * J l R * Aᵀ = J l R := by simp [symplecticGroup] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | mem_iff | null |
coeMatrix : Coe (symplecticGroup l R) (Matrix (l ⊕ l) (l ⊕ l) R) :=
⟨Subtype.val⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | coeMatrix | null |
J_mem : J l R ∈ symplecticGroup l R := by
rw [mem_iff, J, fromBlocks_multiply, fromBlocks_transpose, fromBlocks_multiply]
simp | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | J_mem | null |
symJ : symplecticGroup l R :=
⟨J l R, J_mem l R⟩
variable {l} {R}
@[simp] | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | symJ | The canonical skew-symmetric matrix as an element in the symplectic group. |
coe_J : ↑(symJ l R) = J l R := rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | coe_J | null |
neg_mem (h : A ∈ symplecticGroup l R) : -A ∈ symplecticGroup l R := by
rw [mem_iff] at h ⊢
simp [h] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | neg_mem | null |
symplectic_det (hA : A ∈ symplecticGroup l R) : IsUnit <| det A := by
rw [isUnit_iff_exists_inv]
use A.det
refine (isUnit_det_J l R).mul_left_cancel ?_
rw [mul_one]
rw [mem_iff] at hA
apply_fun det at hA
simp only [det_mul, det_transpose] at hA
rw [mul_comm A.det, mul_assoc] at hA
exact hA | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | symplectic_det | null |
transpose_mem (hA : A ∈ symplecticGroup l R) : Aᵀ ∈ symplecticGroup l R := by
rw [mem_iff] at hA ⊢
rw [transpose_transpose]
have huA := symplectic_det hA
have huAT : IsUnit Aᵀ.det := by
rw [Matrix.det_transpose]
exact huA
calc
Aᵀ * J l R * A = (-Aᵀ) * (J l R)⁻¹ * A := by
rw [J_inv]
simp
_ = (-Aᵀ) * (A * J l R * Aᵀ)⁻¹ * A := by rw [hA]
_ = -(Aᵀ * (Aᵀ⁻¹ * (J l R)⁻¹)) * A⁻¹ * A := by
simp only [Matrix.mul_inv_rev, Matrix.mul_assoc, Matrix.neg_mul]
_ = -(J l R)⁻¹ := by
rw [mul_nonsing_inv_cancel_left _ _ huAT, nonsing_inv_mul_cancel_right _ _ huA]
_ = J l R := by simp [J_inv]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | transpose_mem | null |
transpose_mem_iff : Aᵀ ∈ symplecticGroup l R ↔ A ∈ symplecticGroup l R :=
⟨fun hA => by simpa using transpose_mem hA, transpose_mem⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | transpose_mem_iff | null |
mem_iff' : A ∈ symplecticGroup l R ↔ Aᵀ * J l R * A = J l R := by
rw [← transpose_mem_iff, mem_iff, transpose_transpose] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | mem_iff' | null |
hasInv : Inv (symplecticGroup l R) where
inv A := ⟨(-J l R) * (A : Matrix (l ⊕ l) (l ⊕ l) R)ᵀ * J l R,
mul_mem (mul_mem (neg_mem <| J_mem _ _) <| transpose_mem A.2) <| J_mem _ _⟩ | instance | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | hasInv | null |
coe_inv (A : symplecticGroup l R) : (↑A⁻¹ : Matrix _ _ _) = (-J l R) * (↑A)ᵀ * J l R := rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | coe_inv | null |
inv_left_mul_aux (hA : A ∈ symplecticGroup l R) : -(J l R * Aᵀ * J l R * A) = 1 :=
calc
-(J l R * Aᵀ * J l R * A) = (-J l R) * (Aᵀ * J l R * A) := by
simp only [Matrix.mul_assoc, Matrix.neg_mul]
_ = (-J l R) * J l R := by
rw [mem_iff'] at hA
rw [hA]
_ = (-1 : R) • (J l R * J l R) := by simp only [Matrix.neg_mul, neg_smul, one_smul]
_ = (-1 : R) • (-1 : Matrix _ _ _) := by rw [J_squared]
_ = 1 := by simp only [neg_smul_neg, one_smul] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | inv_left_mul_aux | null |
coe_inv' (A : symplecticGroup l R) : (↑A⁻¹ : Matrix (l ⊕ l) (l ⊕ l) R) = (↑A)⁻¹ := by
refine (coe_inv A).trans (inv_eq_left_inv ?_).symm
simp [inv_left_mul_aux] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | coe_inv' | null |
inv_eq_symplectic_inv (A : Matrix (l ⊕ l) (l ⊕ l) R) (hA : A ∈ symplecticGroup l R) :
A⁻¹ = (-J l R) * Aᵀ * J l R :=
inv_eq_left_inv (by simp only [Matrix.neg_mul, inv_left_mul_aux hA]) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/SymplecticGroup.lean | inv_eq_symplectic_inv | null |
traceAux : (M →ₗ[R] M) →ₗ[R] R :=
Matrix.traceLinearMap ι R R ∘ₗ ↑(LinearMap.toMatrix b b) | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | traceAux | The trace of an endomorphism given a basis. |
traceAux_def (b : Basis ι R M) (f : M →ₗ[R] M) :
traceAux R b f = Matrix.trace (LinearMap.toMatrix b b f) :=
rfl | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | traceAux_def | null |
traceAux_eq : traceAux R b = traceAux R c :=
LinearMap.ext fun f =>
calc
Matrix.trace (LinearMap.toMatrix b b f) =
Matrix.trace (LinearMap.toMatrix b b ((LinearMap.id.comp f).comp LinearMap.id)) := by
rw [LinearMap.id_comp, LinearMap.comp_id]
_ = Matrix.trace (LinearMap.toMatrix c b LinearMap.id * LinearMap.toMatrix c c f *
LinearMap.toMatrix b c LinearMap.id) := by
rw [LinearMap.toMatrix_comp _ c, LinearMap.toMatrix_comp _ c]
_ = Matrix.trace (LinearMap.toMatrix c c f * LinearMap.toMatrix b c LinearMap.id *
LinearMap.toMatrix c b LinearMap.id) := by
rw [Matrix.mul_assoc, Matrix.trace_mul_comm]
_ = Matrix.trace (LinearMap.toMatrix c c ((f.comp LinearMap.id).comp LinearMap.id)) := by
rw [LinearMap.toMatrix_comp _ b, LinearMap.toMatrix_comp _ c]
_ = Matrix.trace (LinearMap.toMatrix c c f) := by rw [LinearMap.comp_id, LinearMap.comp_id]
variable (M) in
open Classical in | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | traceAux_eq | null |
trace : (M →ₗ[R] M) →ₗ[R] R :=
if H : ∃ s : Finset M, Nonempty (Basis s R M) then traceAux R H.choose_spec.some else 0
open Classical in | def | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace | Trace of an endomorphism independent of basis. |
trace_eq_matrix_trace_of_finset {s : Finset M} (b : Basis s R M) (f : M →ₗ[R] M) :
trace R M f = Matrix.trace (LinearMap.toMatrix b b f) := by
have : ∃ s : Finset M, Nonempty (Basis s R M) := ⟨s, ⟨b⟩⟩
rw [trace, dif_pos this, ← traceAux_def]
congr 1
apply traceAux_eq | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_eq_matrix_trace_of_finset | Auxiliary lemma for `trace_eq_matrix_trace`. |
trace_eq_matrix_trace (f : M →ₗ[R] M) :
trace R M f = Matrix.trace (LinearMap.toMatrix b b f) := by
classical
rw [trace_eq_matrix_trace_of_finset R b.reindexFinsetRange, ← traceAux_def, ← traceAux_def,
traceAux_eq R b b.reindexFinsetRange]
variable {R} in
@[simp] theorem _root_.Matrix.trace_toLin_eq (A : Matrix ι ι R) (b : Basis ι R M) :
LinearMap.trace R _ (Matrix.toLin b b A) = A.trace := by
simp [trace_eq_matrix_trace R b]
variable {R} in
@[simp] theorem _root_.Matrix.trace_toLin'_eq (A : Matrix ι ι R) :
LinearMap.trace R _ A.toLin' = A.trace :=
A.trace_toLin_eq (Pi.basisFun R ι) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_eq_matrix_trace | null |
trace_mul_comm (f g : M →ₗ[R] M) : trace R M (f * g) = trace R M (g * f) := by
classical
by_cases H : ∃ s : Finset M, Nonempty (Basis s R M)
· let ⟨s, ⟨b⟩⟩ := H
simp_rw [trace_eq_matrix_trace R b, LinearMap.toMatrix_mul]
apply Matrix.trace_mul_comm
· rw [trace, dif_neg H, LinearMap.zero_apply, LinearMap.zero_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_mul_comm | null |
trace_mul_cycle (f g h : M →ₗ[R] M) :
trace R M (f * g * h) = trace R M (h * f * g) := by
rw [LinearMap.trace_mul_comm, ← mul_assoc] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_mul_cycle | null |
trace_mul_cycle' (f g h : M →ₗ[R] M) :
trace R M (f * (g * h)) = trace R M (h * (f * g)) := by
rw [← mul_assoc, LinearMap.trace_mul_comm] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_mul_cycle' | null |
@[simp]
trace_conj (g : M →ₗ[R] M) (f : (M →ₗ[R] M)ˣ) :
trace R M (↑f * g * ↑f⁻¹) = trace R M g := by
rw [trace_mul_comm]
simp
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_conj | The trace of an endomorphism is invariant under conjugation |
trace_lie {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] (f g : Module.End R M) :
trace R M ⁅f, g⁆ = 0 := by
rw [Ring.lie_def, map_sub, trace_mul_comm]
exact sub_self _ | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_lie | null |
trace_eq_contract_of_basis [Finite ι] (b : Basis ι R M) :
LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M := by
classical
cases nonempty_fintype ι
apply Basis.ext (Basis.tensorProduct (Basis.dualBasis b) b)
rintro ⟨i, j⟩
simp only [Function.comp_apply, Basis.tensorProduct_apply, Basis.coe_dualBasis, coe_comp]
rw [trace_eq_matrix_trace R b, toMatrix_dualTensorHom]
obtain rfl | hij := eq_or_ne i j
· simp
rw [Matrix.trace_single_eq_of_ne j i (1 : R) hij.symm]
simp [hij] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_eq_contract_of_basis | The trace of a linear map correspond to the contraction pairing under the isomorphism
`End(M) ≃ M* ⊗ M` |
trace_eq_contract_of_basis' [Fintype ι] [DecidableEq ι] (b : Basis ι R M) :
LinearMap.trace R M = contractLeft R M ∘ₗ (dualTensorHomEquivOfBasis b).symm.toLinearMap := by
simp [LinearEquiv.eq_comp_toLinearMap_symm, trace_eq_contract_of_basis b] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_eq_contract_of_basis' | The trace of a linear map corresponds to the contraction pairing under the isomorphism
`End(M) ≃ M* ⊗ M`. |
@[simp]
trace_eq_contract : LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M :=
trace_eq_contract_of_basis (Module.Free.chooseBasis R M)
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_eq_contract | When `M` is finite free, the trace of a linear map corresponds to the contraction pairing under
the isomorphism `End(M) ≃ M* ⊗ M`. |
trace_eq_contract_apply (x : Module.Dual R M ⊗[R] M) :
(LinearMap.trace R M) ((dualTensorHom R M M) x) = contractLeft R M x := by
rw [← comp_apply, trace_eq_contract] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_eq_contract_apply | null |
trace_eq_contract' :
LinearMap.trace R M = contractLeft R M ∘ₗ (dualTensorHomEquiv R M M).symm.toLinearMap :=
trace_eq_contract_of_basis' (Module.Free.chooseBasis R M) | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_eq_contract' | When `M` is finite free, the trace of a linear map corresponds to the contraction pairing under
the isomorphism `End(M) ≃ M* ⊗ M`. |
@[simp]
trace_one : trace R M 1 = (finrank R M : R) := by
cases subsingleton_or_nontrivial R
· simp [eq_iff_true_of_subsingleton]
have b := Module.Free.chooseBasis R M
rw [trace_eq_matrix_trace R b, toMatrix_one, finrank_eq_card_chooseBasisIndex]
simp | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_one | The trace of the identity endomorphism is the dimension of the free module. |
@[simp]
trace_id : trace R M id = (finrank R M : R) := by rw [← Module.End.one_eq_id, trace_one]
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_id | The trace of the identity endomorphism is the dimension of the free module. |
trace_transpose : trace R (Module.Dual R M) ∘ₗ Module.Dual.transpose = trace R M := by
let e := dualTensorHomEquiv R M M
have h : Function.Surjective e.toLinearMap := e.surjective
refine (cancel_right h).1 ?_
ext f m; simp [e] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_transpose | null |
trace_prodMap :
trace R (M × N) ∘ₗ prodMapLinear R M N M N R =
(coprod id id : R × R →ₗ[R] R) ∘ₗ prodMap (trace R M) (trace R N) := by
let e := (dualTensorHomEquiv R M M).prodCongr (dualTensorHomEquiv R N N)
have h : Function.Surjective e.toLinearMap := e.surjective
refine (cancel_right h).1 ?_
ext
· simp only [dualTensorHomEquiv, LinearEquiv.coe_prodCongr,
dualTensorHomEquivOfBasis_toLinearMap, AlgebraTensorModule.curry_apply, restrictScalars_comp,
curry_apply, coe_comp, coe_restrictScalars, coe_inl, Function.comp_apply, prodMap_apply,
map_zero, prodMapLinear_apply, dualTensorHom_prodMap_zero, trace_eq_contract_apply,
contractLeft_apply, coe_fst, coprod_apply, id_coe, id_eq, add_zero, e]
· simp only [dualTensorHomEquiv, LinearEquiv.coe_prodCongr,
dualTensorHomEquivOfBasis_toLinearMap, AlgebraTensorModule.curry_apply, restrictScalars_comp,
curry_apply, coe_comp, coe_restrictScalars, coe_inr, Function.comp_apply, prodMap_apply,
map_zero, prodMapLinear_apply, zero_prodMap_dualTensorHom, trace_eq_contract_apply,
contractLeft_apply, coe_snd, coprod_apply, id_coe, id_eq, zero_add, e]
variable {R M N P} | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_prodMap | null |
trace_prodMap' (f : M →ₗ[R] M) (g : N →ₗ[R] N) :
trace R (M × N) (prodMap f g) = trace R M f + trace R N g := by
have h := LinearMap.ext_iff.1 (trace_prodMap R M N) (f, g)
simp only [coe_comp, Function.comp_apply, prodMap_apply, coprod_apply, id,
prodMapLinear_apply] at h
exact h
variable (R M N P)
open TensorProduct Function | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_prodMap' | null |
trace_tensorProduct : compr₂ (mapBilinear R M N M N) (trace R (M ⊗ N)) =
compl₁₂ (lsmul R R : R →ₗ[R] R →ₗ[R] R) (trace R M) (trace R N) := by
apply
(compl₁₂_inj (show Surjective (dualTensorHom R M M) from (dualTensorHomEquiv R M M).surjective)
(show Surjective (dualTensorHom R N N) from (dualTensorHomEquiv R N N).surjective)).1
ext f m g n
simp only [AlgebraTensorModule.curry_apply, TensorProduct.curry_apply,
coe_restrictScalars, compl₁₂_apply, compr₂_apply, mapBilinear_apply,
trace_eq_contract_apply, contractLeft_apply, lsmul_apply, Algebra.id.smul_eq_mul,
map_dualTensorHom, dualDistrib_apply] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_tensorProduct | null |
trace_comp_comm :
compr₂ (llcomp R M N M) (trace R M) = compr₂ (llcomp R N M N).flip (trace R N) := by
apply
(compl₁₂_inj (show Surjective (dualTensorHom R N M) from (dualTensorHomEquiv R N M).surjective)
(show Surjective (dualTensorHom R M N) from (dualTensorHomEquiv R M N).surjective)).1
ext g m f n
simp only [AlgebraTensorModule.curry_apply, TensorProduct.curry_apply,
coe_restrictScalars, compl₁₂_apply, compr₂_apply, flip_apply, llcomp_apply',
comp_dualTensorHom, LinearMapClass.map_smul, trace_eq_contract_apply,
contractLeft_apply, smul_eq_mul, mul_comm]
variable {R M N P}
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_comp_comm | null |
trace_transpose' (f : M →ₗ[R] M) :
trace R _ (Module.Dual.transpose (R := R) f) = trace R M f := by
rw [← comp_apply, trace_transpose] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_transpose' | null |
trace_tensorProduct' (f : M →ₗ[R] M) (g : N →ₗ[R] N) :
trace R (M ⊗ N) (map f g) = trace R M f * trace R N g := by
have h := LinearMap.ext_iff.1 (LinearMap.ext_iff.1 (trace_tensorProduct R M N) f) g
simp only [compr₂_apply, mapBilinear_apply, compl₁₂_apply, lsmul_apply,
Algebra.id.smul_eq_mul] at h
exact h | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_tensorProduct' | null |
trace_comp_comm' (f : M →ₗ[R] N) (g : N →ₗ[R] M) :
trace R M (g ∘ₗ f) = trace R N (f ∘ₗ g) := by
have h := LinearMap.ext_iff.1 (LinearMap.ext_iff.1 (trace_comp_comm R M N) g) f
simp only [llcomp_apply', compr₂_apply, flip_apply] at h
exact h
@[simp] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_comp_comm' | null |
trace_smulRight (f : M →ₗ[R] R) (x : M) :
trace R M (f.smulRight x) = f x := by
rw [trace_eq_matrix_trace _ (Free.chooseBasis R M), ← (Free.chooseBasis R M).sum_repr x]
simp [- Basis.sum_repr, dotProduct] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_smulRight | null |
trace_comp_cycle (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M) :
trace R P (g ∘ₗ f ∘ₗ h) = trace R N (f ∘ₗ h ∘ₗ g) := by
rw [trace_comp_comm', comp_assoc]
variable [Module.Free R M] [Module.Finite R M] [Module.Free R P] [Module.Finite R P] in | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_comp_cycle | null |
trace_comp_cycle' (f : M →ₗ[R] N) (g : N →ₗ[R] P) (h : P →ₗ[R] M) :
trace R P ((g ∘ₗ f) ∘ₗ h) = trace R M ((h ∘ₗ g) ∘ₗ f) := by
rw [trace_comp_comm', ← comp_assoc]
@[simp] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_comp_cycle' | null |
trace_conj' (f : M →ₗ[R] M) (e : M ≃ₗ[R] N) : trace R N (e.conj f) = trace R M f := by
classical
by_cases hM : ∃ s : Finset M, Nonempty (Basis s R M)
· obtain ⟨s, ⟨b⟩⟩ := hM
haveI := Module.Finite.of_basis b
haveI := (Module.free_def R M).mpr ⟨_, ⟨b⟩⟩
haveI := Module.Finite.of_basis (b.map e)
haveI := (Module.free_def R N).mpr ⟨_, ⟨(b.map e).reindex (e.toEquiv.image _)⟩⟩
rw [e.conj_apply, trace_comp_comm', ← comp_assoc, LinearEquiv.comp_coe,
LinearEquiv.self_trans_symm, LinearEquiv.refl_toLinearMap, id_comp]
· rw [trace, trace, dif_neg hM, dif_neg ?_, zero_apply, zero_apply]
rintro ⟨s, ⟨b⟩⟩
exact hM ⟨s.image e.symm, ⟨(b.map e.symm).reindex
((e.symm.toEquiv.image s).trans (Equiv.setCongr Finset.coe_image.symm))⟩⟩ | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_conj' | null |
IsProj.trace {p : Submodule R M} {f : M →ₗ[R] M} (h : IsProj p f) [Module.Free R p]
[Module.Finite R p] [Module.Free R (ker f)] [Module.Finite R (ker f)] :
trace R M f = (finrank R p : R) := by
rw [h.eq_conj_prodMap, trace_conj', trace_prodMap', trace_id, map_zero, add_zero] | theorem | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | IsProj.trace | null |
isNilpotent_trace_of_isNilpotent {f : M →ₗ[R] M} (hf : IsNilpotent f) :
IsNilpotent (trace R M f) := by
by_cases H : ∃ s : Finset M, Nonempty (Basis s R M)
swap
· rw [LinearMap.trace, dif_neg H]
exact IsNilpotent.zero
obtain ⟨s, ⟨b⟩⟩ := H
classical
rw [trace_eq_matrix_trace R b]
apply Matrix.isNilpotent_trace_of_isNilpotent
simpa | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | isNilpotent_trace_of_isNilpotent | null |
trace_comp_eq_mul_of_commute_of_isNilpotent [IsReduced R] {f g : Module.End R M}
(μ : R) (h_comm : Commute f g) (hg : IsNilpotent (g - algebraMap R _ μ)) :
trace R M (f ∘ₗ g) = μ * trace R M f := by
set n := g - algebraMap R _ μ
replace hg : trace R M (f ∘ₗ n) = 0 := by
rw [← isNilpotent_iff_eq_zero, ← Module.End.mul_eq_comp]
refine isNilpotent_trace_of_isNilpotent (Commute.isNilpotent_mul_left ?_ hg)
exact h_comm.sub_right (Algebra.commute_algebraMap_right μ f)
have hμ : g = algebraMap R _ μ + n := eq_add_of_sub_eq' rfl
have : f ∘ₗ algebraMap R _ μ = μ • f := by ext; simp -- TODO Surely exists?
rw [hμ, comp_add, map_add, hg, add_zero, this, LinearMap.map_smul, smul_eq_mul]
@[simp] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_comp_eq_mul_of_commute_of_isNilpotent | null |
trace_baseChange [Module.Free R M] [Module.Finite R M]
(f : M →ₗ[R] M) (A : Type*) [CommRing A] [Algebra R A] :
trace A _ (f.baseChange A) = algebraMap R A (trace R _ f) := by
let b := Module.Free.chooseBasis R M
let b' := Algebra.TensorProduct.basis A b
change _ = (algebraMap R A : R →+ A) _
simp [b', trace_eq_matrix_trace R b, trace_eq_matrix_trace A b', AddMonoidHom.map_trace] | lemma | LinearAlgebra | [
"Mathlib.LinearAlgebra.Contraction",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.RingTheory.Finiteness.Prod",
"Mathlib.RingTheory.TensorProduct.Finite",
"Mathlib.RingTheory.TensorProduct.Free"
] | Mathlib/LinearAlgebra/Trace.lean | trace_baseChange | null |
unitaryGroup : Submonoid (Matrix n n α) :=
unitary (Matrix n n α) | abbrev | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | unitaryGroup | `Matrix.unitaryGroup n` is the group of `n` by `n` matrices where the star-transpose is the
inverse. |
mem_unitaryGroup_iff : A ∈ Matrix.unitaryGroup n α ↔ A * star A = 1 := by
refine ⟨And.right, fun hA => ⟨?_, hA⟩⟩
simpa only [mul_eq_one_comm] using hA | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | mem_unitaryGroup_iff | null |
mem_unitaryGroup_iff' : A ∈ Matrix.unitaryGroup n α ↔ star A * A = 1 := by
refine ⟨And.left, fun hA => ⟨hA, ?_⟩⟩
rwa [mul_eq_one_comm] at hA | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | mem_unitaryGroup_iff' | null |
det_of_mem_unitary {A : Matrix n n α} (hA : A ∈ Matrix.unitaryGroup n α) :
A.det ∈ unitary α := by
constructor
· simpa [star, det_transpose] using congr_arg det hA.1
· simpa [star, det_transpose] using congr_arg det hA.2
open scoped Kronecker in | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | det_of_mem_unitary | null |
kronecker_mem_unitary {R m : Type*} [Semiring R] [StarRing R] [Fintype m]
[DecidableEq m] {U₁ : Matrix n n R} {U₂ : Matrix m m R}
(hU₁ : U₁ ∈ unitary (Matrix n n R)) (hU₂ : U₂ ∈ unitary (Matrix m m R)) :
U₁ ⊗ₖ U₂ ∈ unitary (Matrix (n × m) (n × m) R) := by
simp_rw [unitary.mem_iff, star_eq_conjTranspose, conjTranspose_kronecker']
constructor <;> ext <;> simp only [mul_apply, submatrix_apply, kroneckerMap_apply, Prod.fst_swap,
conjTranspose_apply, ← star_apply, Prod.snd_swap, ← mul_assoc]
· simp_rw [mul_assoc _ (star U₁ _ _), ← Finset.univ_product_univ, Finset.sum_product]
rw [Finset.sum_comm]
simp_rw [← Finset.sum_mul, ← Finset.mul_sum, ← Matrix.mul_apply, hU₁.1, Matrix.one_apply,
mul_boole, ite_mul, zero_mul, Finset.sum_ite_irrel, ← Matrix.mul_apply, hU₂.1,
Matrix.one_apply, Finset.sum_const_zero, ← ite_and, Prod.eq_iff_fst_eq_snd_eq]
· simp_rw [mul_assoc _ _ (star U₂ _ _), ← Finset.univ_product_univ, Finset.sum_product,
← Finset.sum_mul, ← Finset.mul_sum, ← Matrix.mul_apply, hU₂.2, Matrix.one_apply, mul_boole,
ite_mul, zero_mul, Finset.sum_ite_irrel, ← Matrix.mul_apply, hU₁.2, Matrix.one_apply,
Finset.sum_const_zero, ← ite_and, and_comm, Prod.eq_iff_fst_eq_snd_eq] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | kronecker_mem_unitary | The kronecker product of two unitary matrices is unitary.
This is stated for `unitary` instead of `unitaryGroup` as it holds even for
non-commutative coefficients. |
coeMatrix : Coe (unitaryGroup n α) (Matrix n n α) :=
⟨Subtype.val⟩ | instance | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | coeMatrix | null |
coeFun : CoeFun (unitaryGroup n α) fun _ => n → n → α where coe A := A.val | instance | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | coeFun | null |
toLin' (A : unitaryGroup n α) :=
Matrix.toLin' A.1 | def | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | toLin' | `Matrix.UnitaryGroup.toLin' A` is matrix multiplication of vectors by `A`, as a linear map.
After the group structure on `Matrix.unitaryGroup n` is defined, we show in
`Matrix.UnitaryGroup.toLinearEquiv` that this gives a linear equivalence. |
ext_iff (A B : unitaryGroup n α) : A = B ↔ ∀ i j, A i j = B i j :=
Subtype.ext_iff.trans ⟨fun h i j => congr_fun (congr_fun h i) j, Matrix.ext⟩
@[ext] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | ext_iff | null |
ext (A B : unitaryGroup n α) : (∀ i j, A i j = B i j) → A = B :=
(UnitaryGroup.ext_iff A B).mpr | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | ext | null |
star_mul_self (A : unitaryGroup n α) : star A.1 * A.1 = 1 :=
A.2.1
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | star_mul_self | null |
det_isUnit (A : unitaryGroup n α) : IsUnit (A : Matrix n n α).det :=
isUnit_iff_isUnit_det _ |>.mp <| (unitary.toUnits A).isUnit | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | det_isUnit | null |
@[simp] inv_val : ↑A⁻¹ = (star A : Matrix n n α) := rfl
@[simp] theorem inv_apply : ⇑A⁻¹ = (star A : Matrix n n α) := rfl
@[simp] theorem mul_val : ↑(A * B) = A.1 * B.1 := rfl
@[simp] theorem mul_apply : ⇑(A * B) = A.1 * B.1 := rfl
@[simp] theorem one_val : ↑(1 : unitaryGroup n α) = (1 : Matrix n n α) := rfl
@[simp] theorem one_apply : ⇑(1 : unitaryGroup n α) = (1 : Matrix n n α) := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | inv_val | null |
toLin'_mul : toLin' (A * B) = (toLin' A).comp (toLin' B) :=
Matrix.toLin'_mul A.1 B.1
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | toLin'_mul | null |
toLin'_one : toLin' (1 : unitaryGroup n α) = LinearMap.id :=
Matrix.toLin'_one | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | toLin'_one | null |
toLinearEquiv (A : unitaryGroup n α) : (n → α) ≃ₗ[α] n → α :=
{ Matrix.toLin' A.1 with
invFun := toLin' A⁻¹
left_inv := fun x =>
calc
(toLin' A⁻¹).comp (toLin' A) x = (toLin' (A⁻¹ * A)) x := by rw [← toLin'_mul]
_ = x := by rw [inv_mul_cancel, toLin'_one, id_apply]
right_inv := fun x =>
calc
(toLin' A).comp (toLin' A⁻¹) x = toLin' (A * A⁻¹) x := by rw [← toLin'_mul]
_ = x := by rw [mul_inv_cancel, toLin'_one, id_apply] } | def | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | toLinearEquiv | `Matrix.unitaryGroup.toLinearEquiv A` is matrix multiplication of vectors by `A`, as a linear
equivalence. |
toGL (A : unitaryGroup n α) : GeneralLinearGroup α (n → α) :=
GeneralLinearGroup.ofLinearEquiv (toLinearEquiv A) | def | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | toGL | `Matrix.unitaryGroup.toGL` is the map from the unitary group to the general linear group |
coe_toGL (A : unitaryGroup n α) : (toGL A).1 = toLin' A := rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | coe_toGL | null |
toGL_one : toGL (1 : unitaryGroup n α) = 1 := Units.ext <| by
simp only [coe_toGL, toLin'_one]
rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | toGL_one | null |
toGL_mul (A B : unitaryGroup n α) : toGL (A * B) = toGL A * toGL B := Units.ext <| by
simp only [coe_toGL, toLin'_mul]
rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | toGL_mul | null |
embeddingGL : unitaryGroup n α →* GeneralLinearGroup α (n → α) :=
⟨⟨fun A => toGL A, toGL_one⟩, toGL_mul⟩ | def | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | embeddingGL | `Matrix.unitaryGroup.embeddingGL` is the embedding from `Matrix.unitaryGroup n α` to
`LinearMap.GeneralLinearGroup n α`. |
specialUnitaryGroup : Submonoid (Matrix n n α) := unitaryGroup n α ⊓ MonoidHom.mker detMonoidHom
variable {n} {α} | def | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | specialUnitaryGroup | `Matrix.specialUnitaryGroup` is the group of unitary `n` by `n` matrices where the determinant
is 1. (This definition is only correct if 2 is invertible.) |
specialUnitaryGroup_le_unitaryGroup : specialUnitaryGroup n α ≤ unitaryGroup n α :=
inf_le_left | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | specialUnitaryGroup_le_unitaryGroup | null |
mem_specialUnitaryGroup_iff :
A ∈ specialUnitaryGroup n α ↔ A ∈ unitaryGroup n α ∧ A.det = 1 :=
Iff.rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | mem_specialUnitaryGroup_iff | null |
@[simp, norm_cast]
specialUnitaryGroup.coe_star (A : specialUnitaryGroup n α) : (star A).1 = star A.1 := rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | specialUnitaryGroup.coe_star | null |
star_eq_inv (A : specialUnitaryGroup n α) : star A = A⁻¹ :=
rfl | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | star_eq_inv | null |
orthogonalGroup := unitaryGroup n R | abbrev | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | orthogonalGroup | `Matrix.orthogonalGroup n` is the group of `n` by `n` matrices where the transpose is the
inverse. |
mem_orthogonalGroup_iff {A : Matrix n n R} :
A ∈ Matrix.orthogonalGroup n R ↔ A * Aᵀ = 1 :=
mem_unitaryGroup_iff | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | mem_orthogonalGroup_iff | null |
mem_orthogonalGroup_iff' {A : Matrix n n R} :
A ∈ Matrix.orthogonalGroup n R ↔ Aᵀ * A = 1 :=
mem_unitaryGroup_iff' | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | mem_orthogonalGroup_iff' | null |
specialOrthogonalGroup : Submonoid (Matrix n n R) := specialUnitaryGroup n R
variable {n} {R} {A : Matrix n n R} | abbrev | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | specialOrthogonalGroup | `Matrix.specialOrthogonalGroup n` is the group of orthogonal `n` by `n` where the determinant
is one. (This definition is only correct if 2 is invertible.) |
mem_specialOrthogonalGroup_iff :
A ∈ specialOrthogonalGroup n R ↔ A ∈ orthogonalGroup n R ∧ A.det = 1 :=
Iff.rfl
@[simp] | theorem | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | mem_specialOrthogonalGroup_iff | null |
of_mem_specialOrthogonalGroup_fin_two_iff {a b c d : R} :
!![a, b; c, d] ∈ Matrix.specialOrthogonalGroup (Fin 2) R ↔
a = d ∧ b = -c ∧ a ^ 2 + b ^ 2 = 1 := by
trans ((a * a + b * b = 1 ∧ a * c + b * d = 0) ∧
c * a + d * b = 0 ∧ c * c + d * d = 1) ∧ a * d - b * c = 1
· simp [Matrix.mem_specialOrthogonalGroup_iff, Matrix.mem_orthogonalGroup_iff,
← Matrix.ext_iff, Fin.forall_fin_succ, Matrix.vecHead, Matrix.vecTail]
refine ⟨?_, ?_⟩
· rintro ⟨⟨⟨h₀, h₁⟩, -, h₂⟩, h₃⟩
refine ⟨?_, ?_, ?_⟩
· linear_combination - a * h₂ + c * h₁ + d * h₃
· linear_combination - c * h₀ + a * h₁ - b * h₃
· linear_combination h₀
· rintro ⟨rfl, rfl, H⟩
ring_nf at H ⊢
tauto | lemma | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | of_mem_specialOrthogonalGroup_fin_two_iff | null |
mem_specialOrthogonalGroup_fin_two_iff {M : Matrix (Fin 2) (Fin 2) R} :
M ∈ Matrix.specialOrthogonalGroup (Fin 2) R ↔
M 0 0 = M 1 1 ∧ M 0 1 = - M 1 0 ∧ M 0 0 ^ 2 + M 0 1 ^ 2 = 1 := by
rw [← M.etaExpand_eq]
exact of_mem_specialOrthogonalGroup_fin_two_iff | lemma | LinearAlgebra | [
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Data.Matrix.Reflection",
"Mathlib.LinearAlgebra.GeneralLinearGroup",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.LinearAlgebra.Matrix.NonsingularInverse"
] | Mathlib/LinearAlgebra/UnitaryGroup.lean | mem_specialOrthogonalGroup_fin_two_iff | null |
rectVandermonde {α : Type*} (v w : α → R) (n : ℕ) : Matrix α (Fin n) R :=
.of fun i j ↦ (v i) ^ j.1 * (w i) ^ j.rev.1 | def | LinearAlgebra | [
"Mathlib.Data.Nat.Factorial.SuperFactorial",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.RingTheory.Polynomial.Pochhammer"
] | Mathlib/LinearAlgebra/Vandermonde.lean | rectVandermonde | A matrix with rows all having the form `[b^(n-1), a * b^(n-2), ..., a ^ (n-1)]` |
projVandermonde (v w : Fin n → R) : Matrix (Fin n) (Fin n) R :=
rectVandermonde v w n | def | LinearAlgebra | [
"Mathlib.Data.Nat.Factorial.SuperFactorial",
"Mathlib.LinearAlgebra.Matrix.Block",
"Mathlib.LinearAlgebra.Matrix.Nondegenerate",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.RingTheory.Polynomial.Pochhammer"
] | Mathlib/LinearAlgebra/Vandermonde.lean | projVandermonde | A square matrix with rows all having the form `[b^(n-1), a * b^(n-2), ..., a ^ (n-1)]` |
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