File size: 3,021 Bytes
8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import Mathlib
import OAI.Geometry.Borsuk.MatrixGeometry.MatrixTransport
import OAI.Geometry.Borsuk.MatrixGeometry.ReflectionTransport
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator
section MatrixReflection
variable {ν : Type*} [Fintype ν] [DecidableEq ν]
local notation "E" => EuclideanSpace ℝ ν
local notation "M" => Matrix ν ν ℝ
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)
omit [Fintype ν] [DecidableEq ν] in
lemma rankOne_continuous : Continuous (rankOne (ν := ν)) := by
apply continuous_matrix
intro i j
exact (PiLp.continuous_apply 2 (fun _ : ν => ℝ) i).mul
(PiLp.continuous_apply 2 (fun _ : ν => ℝ) j)
noncomputable def reflectionMatrix (c u : E) : M :=
(2 / ‖c+u‖^2) • rankOne (c+u) - 1
lemma reflectionMatrix_action (c u v : E) :
op (reflectionMatrix c u) v = householder c u v := by
rw [reflectionMatrix,map_sub,map_smul,map_one,sub_apply,
smul_apply,one_apply_eq_self,rankOne_action,householder_apply,
smul_smul]
congr 2
ring
lemma reflectionMatrix_eq (c u : E) :
op (reflectionMatrix c u) = (householder c u).toContinuousLinearEquiv.toContinuousLinearMap := by
apply ContinuousLinearMap.ext
intro v
exact reflectionMatrix_action c u v
omit [DecidableEq ν] in
lemma householder_involutive (c u : E) : Function.Involutive (householder c u) :=
(Submodule.span ℝ {c+u}).reflection_involutive
lemma reflectionMatrix_conjugate (c u : E) (B : M) :
matrixConjugate (householder c u) B = reflectionMatrix c u * B * reflectionMatrix c u := by
apply (Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)).injective
change op (matrixConjugate (householder c u) B) = op (reflectionMatrix c u * B * reflectionMatrix c u)
rw [map_mul,map_mul]
apply ContinuousLinearMap.ext
intro v
rw [matrixConjugate_action]
change _ = op (reflectionMatrix c u) (op B (op (reflectionMatrix c u) v))
rw [reflectionMatrix_action,reflectionMatrix_action]
congr 2
lemma reflectionMatrix_continuousOn (c : E) :
ContinuousOn (reflectionMatrix c) {u | c+u ≠ 0} := by
have hn : Continuous (fun u : E => ‖c+u‖^2) := (continuous_const.add continuous_id).norm.pow 2
have hd : ContinuousOn (fun u : E => 2 / ‖c+u‖^2) {u | c+u ≠ 0} :=
continuous_const.continuousOn.div hn.continuousOn
(fun u hu => pow_ne_zero 2 (norm_ne_zero_iff.mpr hu))
exact (hd.smul (rankOne_continuous.comp (continuous_const.add continuous_id)).continuousOn).sub
continuous_const.continuousOn
lemma householder_matrix_joint_continuousOn (c : E) :
ContinuousOn (fun p : E × M => matrixConjugate (householder c p.1) p.2)
{p | c+p.1 ≠ 0} := by
simp_rw [reflectionMatrix_conjugate]
have hR : ContinuousOn (fun p : E × M => reflectionMatrix c p.1) {p | c+p.1 ≠ 0} :=
(reflectionMatrix_continuousOn c).comp continuous_fst.continuousOn (fun _ h => h)
exact (hR.mul continuous_snd.continuousOn).mul hR
end MatrixReflection
end BorsukNine
end OAI
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