| import Mathlib |
| import OAI.Geometry.Borsuk.Support.DisjointChambers |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped Matrix RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator |
| section MatrixCompression |
| variable {μ ν : Type*} [Fintype μ] [Fintype ν] [DecidableEq μ] [DecidableEq ν] |
| local notation "D" => EuclideanSpace ℝ μ |
| local notation "E" => EuclideanSpace ℝ ν |
| |
| noncomputable def matrixCompress (e : D →ₗᵢ[ℝ] E) : Matrix ν ν ℝ →ₗ[ℝ] Matrix μ μ ℝ where |
| toFun Q := (isometryMatrix e)ᴴ * Q * isometryMatrix e |
| map_add' Q R := by simp only [Matrix.mul_add,Matrix.add_mul] |
| map_smul' t Q := by simp only [Matrix.mul_smul,Matrix.smul_mul]; rfl |
| |
| omit [DecidableEq ν] in |
| lemma matrixCompress_nonneg (e : D →ₗᵢ[ℝ] E) {Q : Matrix ν ν ℝ} (hQ : 0 ≤ Q) : |
| 0 ≤ matrixCompress e Q := |
| ((Matrix.nonneg_iff_posSemidef.mp hQ).conjTranspose_mul_mul_same _).nonneg |
|
|
| lemma isometryMatrix_adjoint_action (e : D →ₗᵢ[ℝ] E) (x : D) : |
| Matrix.toEuclideanLin (isometryMatrix e)ᴴ (e x) = x := by |
| rw [← isometryMatrix_action e x] |
| change ((Matrix.toEuclideanLin (isometryMatrix e)ᴴ).comp |
| (Matrix.toEuclideanLin (isometryMatrix e))) x = x |
| rw [← toEuclideanLin_mul,isometryMatrix_star_mul] |
| simp only [Matrix.toLpLin_apply,Matrix.one_mulVec,WithLp.toLp_ofLp] |
|
|
| lemma matrixEmbed_compress (e : D →ₗᵢ[ℝ] E) {Q : Matrix ν ν ℝ} |
| (hQ : IsSelfAdjoint Q) |
| (hR : LinearMap.range (Matrix.toEuclideanLin Q) ≤ LinearMap.range e.toLinearMap) : |
| matrixEmbed e (matrixCompress e Q) = Q := by |
| let T := isometryMatrix e |
| have hleft : (T * Tᴴ) * Q = Q := by |
| apply Matrix.toEuclideanLin.injective |
| apply LinearMap.ext |
| intro v |
| obtain ⟨x,hx⟩ := hR (LinearMap.mem_range_self (Matrix.toEuclideanLin Q) v) |
| change e x = Matrix.toEuclideanLin Q v at hx |
| rw [toEuclideanLin_mul,toEuclideanLin_mul] |
| change Matrix.toEuclideanLin T (Matrix.toEuclideanLin Tᴴ (Matrix.toEuclideanLin Q v)) = _ |
| rw [← hx,isometryMatrix_adjoint_action,isometryMatrix_action] |
| have hright : Q * (T*Tᴴ) = Q := by |
| have he := congrArg Matrix.conjTranspose hleft |
| rw [Matrix.conjTranspose_mul,Matrix.conjTranspose_mul,Matrix.conjTranspose_conjTranspose] at he |
| simpa only [← Matrix.star_eq_conjTranspose,hQ.star_eq] using he |
| change T * (Tᴴ * Q * T) * Tᴴ = Q |
| calc |
| _ = ((T*Tᴴ)*Q)*(T*Tᴴ) := by simp only [Matrix.mul_assoc] |
| _ = Q := by rw [hleft,hright] |
|
|
| lemma matrixCompress_trace (e : D →ₗᵢ[ℝ] E) {Q : Matrix ν ν ℝ} |
| (hQ : IsSelfAdjoint Q) |
| (hR : LinearMap.range (Matrix.toEuclideanLin Q) ≤ LinearMap.range e.toLinearMap) : |
| Matrix.trace (matrixCompress e Q) = Matrix.trace Q := by |
| rw [← matrixEmbed_trace e,matrixEmbed_compress e hQ hR] |
|
|
| end MatrixCompression |
|
|
| end BorsukNine |
|
|
| end OAI |
|
|
