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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 | import Mathlib
import OAI.Geometry.Borsuk.MatrixGeometry.Rank
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator
section MatrixTransport
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)
noncomputable def matrixConjugate (e : E ≃ₗᵢ[ℝ] E) : M ≃⋆ₐ[ℝ] M :=
(op).trans (e.conjStarAlgEquiv.trans (op).symm)
lemma matrixConjugate_action (e : E ≃ₗᵢ[ℝ] E) (A : M) (v : E) :
op (matrixConjugate e A) v = e (op A (e.symm v)) := by
simp only [matrixConjugate,StarAlgEquiv.trans_apply,StarAlgEquiv.apply_symm_apply,
LinearIsometryEquiv.conjStarAlgEquiv_apply_apply]
lemma matrixConjugate_nonneg (e : E ≃ₗᵢ[ℝ] E) {A : M} (hA : 0 ≤ A) :
0 ≤ matrixConjugate e A := map_nonneg _ hA
lemma matrixConjugate_rankOne (e : E ≃ₗᵢ[ℝ] E) (u : E) :
matrixConjugate e (rankOne u) = rankOne (e u) := by
apply (Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)).injective
change op (matrixConjugate e (rankOne u)) = op (rankOne (e u))
apply ContinuousLinearMap.ext
intro v
rw [matrixConjugate_action,rankOne_action,rankOne_action,map_smul]
congr 1
simpa using (e.inner_map_map u (e.symm v)).symm
lemma matrix_operator_trace (A : M) : LinearMap.trace ℝ E (op A).toLinearMap = Matrix.trace A := by
rw [Matrix.coe_toEuclideanCLM_eq_toEuclideanLin,Matrix.toEuclideanLin_eq_toLin_orthonormal]
exact Matrix.trace_toLin_eq A _
lemma matrixConjugate_trace (e : E ≃ₗᵢ[ℝ] E) (A : M) :
Matrix.trace (matrixConjugate e A) = Matrix.trace A := by
rw [← matrix_operator_trace,← matrix_operator_trace]
have he : (op (matrixConjugate e A)).toLinearMap = e.toLinearEquiv.conj (op A).toLinearMap := by
apply LinearMap.ext
intro v
exact matrixConjugate_action e A v
rw [he,LinearMap.trace_conj']
lemma matrixConjugate_kernel (e : E ≃ₗᵢ[ℝ] E) {A : M} {v : E} (hv : op A v = 0) :
op (matrixConjugate e A) (e v) = 0 := by
rw [matrixConjugate_action,e.symm_apply_apply,hv,map_zero]
lemma matrixConjugate_range (e : E ≃ₗᵢ[ℝ] E) {A : M} {v : E}
(hv : v ∈ LinearMap.range (op A).toLinearMap) :
e v ∈ LinearMap.range (op (matrixConjugate e A)).toLinearMap := by
obtain ⟨x,rfl⟩ := hv
exact ⟨e x,by simp only [ContinuousLinearMap.coe_coe,matrixConjugate_action,e.symm_apply_apply]⟩
lemma matrixConjugate_selfAdjoint (e : E ≃ₗᵢ[ℝ] E) {A : M} (hA : IsSelfAdjoint A) :
IsSelfAdjoint (matrixConjugate e A) := by
rw [IsSelfAdjoint,← map_star,hA.star_eq]
lemma matrixConjugate_involutive {e : E ≃ₗᵢ[ℝ] E}
(he : ∀ v, e (e v) = v) (A : M) : matrixConjugate e (matrixConjugate e A) = A := by
have hes : ∀ v, e.symm v = e v := by intro v; apply e.injective; simp [he]
apply (Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)).injective
change op (matrixConjugate e (matrixConjugate e A)) = op A
apply ContinuousLinearMap.ext
intro v
simp only [matrixConjugate_action,hes,he]
end MatrixTransport
end BorsukNine
end OAI
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