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chore: remove redundant lemma LinearMap.IsSymmetric.eigenspace_invariant_of_commute #16915

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chore: remove redundant lemma LinearMap.IsSymmetric.eigenspace_invariant_of_commute #16915

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11 changes: 2 additions & 9 deletions Mathlib/Analysis/InnerProductSpace/JointEigenspace.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -45,13 +45,6 @@ section Pair

variable {α : 𝕜} {A B : E →ₗ[𝕜] E}

/--If a pair of operators commute, then the eigenspaces of one are invariant under the other.-/
theorem eigenspace_invariant_of_commute
(hAB : A ∘ₗ B = B ∘ₗ A) (α : 𝕜) : ∀ v ∈ (eigenspace A α), (B v ∈ eigenspace A α) := by
intro v hv
rw [eigenspace, mem_ker, sub_apply, Module.algebraMap_end_apply, ← comp_apply A B v, hAB,
comp_apply B A v, ← map_smul, ← map_sub, hv, map_zero] at *

/--The simultaneous eigenspaces of a pair of commuting symmetric operators form an
`OrthogonalFamily`.-/
theorem orthogonalFamily_eigenspace_inf_eigenspace (hA : A.IsSymmetric) (hB : B.IsSymmetric) :
Expand All @@ -71,7 +64,7 @@ operators commute. -/
theorem eigenspace_inf_eigenspace
(hAB : A ∘ₗ B = B ∘ₗ A) (γ : 𝕜) :
eigenspace A α ⊓ eigenspace B γ = map (Submodule.subtype (eigenspace A α))
(eigenspace (B.restrict (eigenspace_invariant_of_commute hAB α)) γ) :=
(eigenspace (B.restrict (mapsTo_genEigenspace_of_comm hAB α 1)) γ) :=
(eigenspace A α).inf_genEigenspace _ _ (k := 1)

variable [FiniteDimensional 𝕜 E]
Expand All @@ -86,7 +79,7 @@ theorem iSup_eigenspace_inf_eigenspace (hB : B.IsSymmetric)
congr 1
rw [← Submodule.orthogonal_eq_bot_iff]
exact orthogonalComplement_iSup_eigenspaces_eq_bot <|
hB.restrict_invariant <| eigenspace_invariant_of_commute hAB α
hB.restrict_invariant <| mapsTo_genEigenspace_of_comm hAB α 1

/-- If A and B are commuting symmetric operators acting on a finite dimensional inner product space,
then the simultaneous eigenspaces of A and B exhaust the space. -/
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