Matrix isometry preserves norm
Matrix.isometry_preserves_norm
Project documentation
The operator norm of a matrix, with respect to the Euclidean norm (l2 norm) on the domain and codomain. -/ noncomputable def Matrix.opNorm [RCLike π] (A : Matrix m n π) : β := βLinearMap.toContinuousLinearMap (Matrix.toEuclideanLin A)β /- An isometry preserves the Euclidean norm.
Exact Lean statement
theorem Matrix.isometry_preserves_norm (A : Matrix n m π) (hA : A.Isometry) (x : EuclideanSpace π m) :
βMatrix.toEuclideanLin A xβ = βxβFormal artifact
Lean source
theorem Matrix.isometry_preserves_norm (A : Matrix n m π) (hA : A.Isometry) (x : EuclideanSpace π m) : βMatrix.toEuclideanLin A xβ = βxβ := by rw [ β sq_eq_sqβ ( by positivity ) ( by positivity ), Matrix.Isometry ] at *; simp [ EuclideanSpace.norm_eq] have h_inner : β x y : EuclideanSpace π m, inner π (toEuclideanLin A x) (toEuclideanLin A y) = inner π x y := by intro x y have h_inner_eq : inner π (toEuclideanLin A x) (toEuclideanLin A y) = inner π x (toEuclideanLin A.conjTranspose (toEuclideanLin A y)) := by simp [ Matrix.toEuclideanLin, inner ]; simp [ Matrix.mulVec, dotProduct, Finset.mul_sum, mul_comm, ]; simp [ Matrix.mul_apply, mul_assoc, mul_comm, mul_left_comm, Finset.mul_sum] rw [Finset.sum_comm, Finset.sum_comm_cycle] simp_all [ Matrix.toEuclideanLin, toLpLin]; convert congr_arg Real.sqrt ( congr_arg ( fun z => βzβ ) ( h_inner x x ) ) using 1; Β· simp [ EuclideanSpace.norm_eq, inner_self_eq_norm_sq_to_K ]; Β· simp [ EuclideanSpace.norm_eq, inner_self_eq_norm_sq_to_K ]- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/SSA.lean:38-52
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Related declarations
Conj Transpose isometry mul isometry le one
conjTranspose_isometry_mul_isometry_le_one
Project documentation
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 π) : Matrix.opNorm Aα΄΄ = Matrix.opNorm A := by unfold Matrix.opNorm rw [β ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...
Source project: quantumInfo
Person-level attribution pending.
Convex roof of pure
convex_roof_of_pure
Plain-language statement
The convex roof extension of g : KetUpToPhase d β ββ₯0 applied to a pure state Ο is g (KetUpToPhase.mk Ο).
Source project: quantumInfo
Person-level attribution pending.
Id achieves Rate log dim
CPTPMap.id_achievesRate_log_dim
Plain-language statement
The identity channel on D dimensional space achieves a rate of log2(D).
Source project: quantumInfo
Person-level attribution pending.