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[Merged by Bors] - chore(Analysis/InnerProductSpace): fix eigenvalue_pos_of_pos #42929

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[Merged by Bors] - chore(Analysis/InnerProductSpace): fix eigenvalue_pos_of_pos #42929

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a940e74
fix lemma
mcdoll Aug 19, 2026
f24777c
unnecessary type annotations
mcdoll Aug 19, 2026
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10 changes: 5 additions & 5 deletions Mathlib/Analysis/InnerProductSpace/Spectrum.lean
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Expand Up @@ -407,22 +407,22 @@ theorem inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜]
simp only [h, inner_smul_right, inner_self_eq_norm_sq_to_K]

theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
(hnn : ∀ x : E, 0 ≤ RCLike.re ⟪x, T x⟫) : 0 ≤ μ := by
(hnn : ∀ x, 0 ≤ RCLike.re ⟪x, T x⟫) : 0 ≤ μ := by
obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_hasEigenvector
have hpos : (0 : ℝ) < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv₂
have hpos : 0 < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv₂
simp only [mem_genEigenspace_one] at hv₁
have : RCLike.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 :=
mod_cast congr_arg RCLike.re (inner_product_apply_eigenvector hv₁)
exact (mul_nonneg_iff_of_pos_right hpos).mp (this ▸ hnn v)

theorem eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
(hnn : ∀ x : E, 0 < RCLike.re ⟪x, T x⟫) : 0 < μ := by
(hnn : ∀ x, x ≠ 0 → 0 < RCLike.re ⟪x, T x⟫) : 0 < μ := by
obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_hasEigenvector
have hpos : (0 : ℝ) < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv₂
have hpos : 0 < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv₂
simp only [mem_genEigenspace_one] at hv₁
have : RCLike.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 :=
mod_cast congr_arg RCLike.re (inner_product_apply_eigenvector hv₁)
exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v)
exact (mul_pos_iff_of_pos_right hpos).mp (this ▸ hnn v hv₂)

end Nonneg

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