| import Mathlib |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| section ReflectionTransport |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| noncomputable def householder (c x : E) : E ≃ₗᵢ[ℝ] E := |
| (Submodule.span ℝ {c+x}).reflection |
|
|
| lemma householder_apply (c x v : E) : |
| householder c x v = (2 * (⟪c+x,v⟫ / ‖c+x‖ ^ 2)) • (c+x) - v := by |
| rw [householder,Submodule.reflection_singleton_apply] |
| simp only [RCLike.ofReal_real_eq_id, id_eq, two_smul, two_mul, add_smul] |
|
|
| lemma householder_send {c x : E} (h : ‖c‖ = ‖x‖) : householder c x c = x := by |
| have hh := Submodule.reflection_sub (v := c) (w := -x) (by simpa using h) |
| rw [sub_neg_eq_add,Submodule.reflection_orthogonal_apply] at hh |
| exact neg_injective hh |
|
|
| lemma householder_orthogonal {c x v : E} (h : ‖c‖ = ‖x‖) (hv : ⟪c,v⟫ = 0) : |
| ⟪x,householder c x v⟫ = 0 := by |
| calc |
| ⟪x,householder c x v⟫ = ⟪householder c x c,householder c x v⟫ := by rw [householder_send h] |
| _ = ⟪c,v⟫ := (householder c x).inner_map_map c v |
| _ = 0 := hv |
|
|
| lemma householder_at_self {c v : E} (hv : ⟪c,v⟫ = 0) : |
| householder c c v = -v := by |
| rw [householder_apply] |
| simp [inner_add_left,hv] |
|
|
| lemma householder_joint_continuousOn (c : E) : |
| ContinuousOn (fun p : E × E => householder c p.1 p.2) {p | c+p.1 ≠ 0} := by |
| have hc : Continuous (fun p : E × E => c+p.1) := continuous_const.add continuous_fst |
| have hz : Continuous (fun p : E × E => p.2) := continuous_snd |
| have hn : Continuous (fun p : E × E => ⟪c+p.1,p.2⟫) := hc.inner hz |
| have hd : Continuous (fun p : E × E => ‖c+p.1‖ ^ 2) := hc.norm.pow 2 |
| have hq : ContinuousOn (fun p : E × E => ⟪c+p.1,p.2⟫ / ‖c+p.1‖ ^ 2) {p | c+p.1 ≠ 0} := |
| hn.continuousOn.div hd.continuousOn (fun _ hp => pow_ne_zero 2 (norm_ne_zero_iff.mpr hp)) |
| have hmul : ContinuousOn (fun p : E × E => (2 : ℝ) * (⟪c+p.1,p.2⟫ / ‖c+p.1‖ ^ 2)) |
| {p | c+p.1 ≠ 0} := continuous_const.continuousOn.mul hq |
| have hs : ContinuousOn (fun p : E × E => ((2 : ℝ) * (⟪c+p.1,p.2⟫ / ‖c+p.1‖ ^ 2)) • (c+p.1) - p.2) |
| {p | c+p.1 ≠ 0} := (hmul.smul hc.continuousOn).sub hz.continuousOn |
| simpa only [householder_apply] using hs |
|
|
| lemma householder_joint_contDiffOn (c : E) : |
| ContDiffOn ℝ (⊤ : ℕ∞) (fun p : E × E => householder c p.1 p.2) {p | c+p.1 ≠ 0} := by |
| have he : (fun p : E × E => ‖c+p.1‖ ^ 2) = fun p => ⟪c+p.1,c+p.1⟫ := by |
| ext p |
| exact (real_inner_self_eq_norm_sq _).symm |
| simp_rw [householder_apply] |
| apply ContDiffOn.sub _ contDiff_snd.contDiffOn |
| apply ContDiffOn.smul _ (contDiff_const.add contDiff_fst).contDiffOn |
| apply contDiff_const.contDiffOn.mul |
| apply ContDiffOn.div |
| (((contDiff_const.add contDiff_fst).inner ℝ contDiff_snd).contDiffOn) |
| (by rw [he]; exact ((contDiff_const.add contDiff_fst).inner ℝ |
| (contDiff_const.add contDiff_fst)).contDiffOn) |
| intro p hp |
| exact pow_ne_zero 2 (norm_ne_zero_iff.mpr hp) |
|
|
| end ReflectionTransport |
| end BorsukNine |
|
|
| end OAI |
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|
