File size: 2,821 Bytes
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 | 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
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