Research proof index

The operator norm of the conjugate transpose is equal to the operator norm. -/ theorem Matrix.opNorm_conjTranspose_eq_opNorm {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n 𝕜) : Matrix.opNorm Aᴴ = Matrix.opNorm A := by unfold Matrix.opNorm rw [← ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...

The operator norm of the conjugate transpose is equal to the operator norm. -/ theorem Matrix.opNorm_conjTranspose_eq_opNorm {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n 𝕜) : Matrix.opNorm Aᴴ = Matrix.opNorm A := by unfold Matrix.opNorm rw [← ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...

The intermediate operator inequality: ρ_AB ⊗ σ_C⁻¹ ≤ (ρ_A ⊗ σ_BC⁻¹).reindex(assoc⁻¹). This is derived from W_mat_sq_le_one by algebraic manipulation (conjugation and simplification).

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.