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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 | import Mathlib
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace ContDiff
open scoped Manifold
open MeasureTheory Set Metric
open scoped ENNReal Pointwise Matrix
section MatrixNorm
open scoped MatrixOrder Matrix.Norms.L2Operator
variable {ν : Type*} [Fintype ν] [DecidableEq ν]
local notation "E" => EuclideanSpace ℝ ν
lemma matrix_action_norm_sq (A : Matrix ν ν ℝ) (x : E) :
‖(Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) A x‖ ^ 2 =
⟪x, (Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) (star A * A) x⟫ := by
rw [map_mul, map_star]
change ‖(Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) A x‖ ^ 2 =
⟪x, (star ((Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) A)) ((Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) A x)⟫
rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right,
real_inner_self_eq_norm_sq]
lemma sqrt_action_norm_sq (Q : Matrix ν ν ℝ) (hQ : 0 ≤ Q) (x : E) :
‖(Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) (CFC.sqrt Q) x‖ ^ 2 = ⟪x, (Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) Q x⟫ := by
rw [matrix_action_norm_sq, (CFC.sqrt_nonneg Q).star_eq, CFC.sqrt_mul_sqrt_self Q hQ]
lemma matrix_sqrt_smul {Q : Matrix ν ν ℝ} (hQ : 0 ≤ Q) {t : ℝ} (ht : 0 ≤ t) :
CFC.sqrt (t • Q) = Real.sqrt t • CFC.sqrt Q := by
apply (CFC.sqrt_eq_iff _ _ (smul_nonneg ht hQ)
(smul_nonneg (Real.sqrt_nonneg _) (CFC.sqrt_nonneg _))).mpr
rw [smul_mul_smul, CFC.sqrt_mul_sqrt_self Q hQ, Real.mul_self_sqrt ht]
end MatrixNorm
section MatrixRange
open scoped MatrixOrder Matrix.Norms.L2Operator
variable {ν : Type*} [Fintype ν] [DecidableEq ν]
local notation "E" => EuclideanSpace ℝ ν
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)
lemma matrix_sqrt_range (Q : Matrix ν ν ℝ) (hQ : 0 ≤ Q) :
LinearMap.range (op (CFC.sqrt Q)).toLinearMap = LinearMap.range (op Q).toLinearMap := by
have hs : star (op (CFC.sqrt Q)) = op (CFC.sqrt Q) := by
rw [← map_star, (CFC.sqrt_nonneg Q).star_eq]
have hh := LinearMap.range_adjoint_comp_self (op (CFC.sqrt Q)).toLinearMap
have ha : (op (CFC.sqrt Q)).toLinearMap.adjoint = (op (CFC.sqrt Q)).toLinearMap := by
change (ContinuousLinearMap.adjoint (op (CFC.sqrt Q))).toLinearMap = _
rw [← ContinuousLinearMap.star_eq_adjoint, hs]
rw [ha] at hh
have hm : (op (CFC.sqrt Q)).toLinearMap.comp (op (CFC.sqrt Q)).toLinearMap = (op Q).toLinearMap := by
change (op (CFC.sqrt Q) * op (CFC.sqrt Q)).toLinearMap = _
rw [← map_mul, CFC.sqrt_mul_sqrt_self Q hQ]
rw [hm] at hh
exact hh.symm
end MatrixRange
end BorsukNine
end OAI
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