| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical ComplexOrder |
| open Matrix |
|
|
| namespace BinaryCoordinateSweeps.Density |
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| def projectionMatrix (S : Submodule ℂ (EuclideanSpace ℂ I)) : Matrix I I ℂ := |
| (Matrix.toEuclideanCLM (𝕜 := ℂ)).symm S.starProjection |
|
|
| lemma projectionMatrix_linear (S : Submodule ℂ (EuclideanSpace ℂ I)) : |
| (projectionMatrix S).toEuclideanLin = S.starProjection.toLinearMap := by |
| have h := (Matrix.toEuclideanCLM (𝕜 := ℂ)).apply_symm_apply S.starProjection |
| exact congrArg ContinuousLinearMap.toLinearMap h |
|
|
| lemma projectionMatrix_psd (S : Submodule ℂ (EuclideanSpace ℂ I)) : |
| (projectionMatrix S).PosSemidef := by |
| rw [← Matrix.isPositive_toEuclideanLin_iff, projectionMatrix_linear] |
| exact (Submodule.isSymmetricProjection_starProjection S).isPositive |
|
|
| lemma projectionMatrix_idempotent (S : Submodule ℂ (EuclideanSpace ℂ I)) : |
| projectionMatrix S * projectionMatrix S = projectionMatrix S := by |
| apply (Matrix.toEuclideanCLM (𝕜 := ℂ)).injective |
| simp only [map_mul] |
| have h : (Matrix.toEuclideanCLM (𝕜 := ℂ)).toRingEquiv (projectionMatrix S) = |
| S.starProjection := (Matrix.toEuclideanCLM (𝕜 := ℂ)).apply_symm_apply _ |
| rw [h] |
| exact S.isIdempotentElem_starProjection |
|
|
| lemma trace_toEuclideanLinear (M : Matrix I I ℂ) : |
| LinearMap.trace ℂ (EuclideanSpace ℂ I) M.toEuclideanLin = M.trace := by |
| rw [LinearMap.trace_eq_sum_inner _ (EuclideanSpace.basisFun I ℂ)] |
| unfold Matrix.trace |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp [EuclideanSpace.basisFun_apply, PiLp.inner_apply, Matrix.ofLp_toLpLin, |
| Matrix.toLin'_apply, PiLp.ofLp_single, Matrix.mulVec, dotProduct, Pi.single_apply] |
|
|
| lemma projectionMatrix_trace (S : Submodule ℂ (EuclideanSpace ℂ I)) : |
| (projectionMatrix S).trace = (Module.finrank ℂ S : ℂ) := by |
| rw [← trace_toEuclideanLinear, projectionMatrix_linear] |
| have h : LinearMap.IsProj S S.starProjection.toLinearMap := |
| ⟨S.starProjection_apply_mem, fun v hv => Submodule.starProjection_eq_self_iff.mpr hv⟩ |
| exact h.trace |
|
|
| lemma projectionMatrix_coeff (S : Submodule ℂ (EuclideanSpace ℂ I)) : |
| LinearMap.toMatrix' ((WithLp.linearEquiv 2 ℂ (I → ℂ)).conj S.starProjection.toLinearMap) = |
| projectionMatrix S := by |
| apply Matrix.toEuclideanLin.injective |
| rw [projectionMatrix_linear] |
| ext v i |
| simp only [Matrix.toEuclideanLin, Matrix.toLpLin, LinearEquiv.conj_apply] |
| simp |
|
|
| omit [DecidableEq I] in |
| lemma projectionMatrix_commutes (S : Submodule ℂ (EuclideanSpace ℂ I)) |
| (f : EuclideanSpace ℂ I →ₗᵢ[ℂ] EuclideanSpace ℂ I) (hf : S.map f.toLinearMap = S) |
| (v : EuclideanSpace ℂ I) : |
| f (S.starProjection v) = S.starProjection (f v) := by |
| have h := f.map_starProjection S v |
| simpa only [hf] using h |
|
|
| end BinaryCoordinateSweeps.Density |
|
|
| end |
|
|
| end OAI |
|
|
