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