| 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 | |