| 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 |
|
|
