File size: 2,924 Bytes
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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
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