Research proof index

The operator norm of a matrix, with respect to the Euclidean norm (l2 norm) on the domain and codomain. -/ noncomputable def Matrix.opNorm [RCLike 𝕜] (A : Matrix m n 𝕜) : ℝ := ‖LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin A)‖ /- An isometry preserves the Euclidean norm.

The operator norm of a matrix, with respect to the Euclidean norm (l2 norm) on the domain and codomain. -/ noncomputable def Matrix.opNorm [RCLike 𝕜] (A : Matrix m n 𝕜) : ℝ := ‖LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin A)‖ /- An isometry preserves the Euclidean norm. -/ theorem Matrix.isometry_preserves_norm (A : Matrix n m 𝕜) (hA : A.Isom...

The operator norm of a matrix product is at most the product of the operator norms.

The Choi matrix of a map in symmetric Kraus form is a sum of rank-1 projectors. -/ theorem choi_of_kraus_R [DecidableEq A] (K : κ → Matrix B A 𝕜) : (of_kraus K K).choi_matrix = ∑ k, Matrix.vecMulVec (fun (x : B × A) => K k x.1 x.2) (fun (x : B × A) => star (K k x.1 x.2)) := by unfold of_kraus choi_matrix ext i j : 2 simp only [LinearMap.coe_sum, LinearMa...

Proves that MatrixMap.choi_matrix and MatrixMap.of_choi_matrix inverses.

The linear map of conjugating a matrix by another, x → y * x * yᴴ. -/ @[simps] def root.MatrixMap.conj (y : Matrix B A R) : MatrixMap A B R where toFun x := y * x * y.conjTranspose map_add' x y := by rw [Matrix.mul_add, Matrix.add_mul] map_smul' r x := by rw [RingHom.id_apply, Matrix.mul_smul, Matrix.smul_mul] omit [DecidableEq A] in theorem conj_isPo...