Matrices as a normed space

In this file we provide the following non-instances for norms on matrices:

• The elementwise norm:

  • Matrix.seminormedAddCommGroup
  • Matrix.normedAddCommGroup
  • Matrix.normedSpace

• The Frobenius norm:

  • Matrix.frobeniusSeminormedAddCommGroup
  • Matrix.frobeniusNormedAddCommGroup
  • Matrix.frobeniusNormedSpace
  • Matrix.frobeniusNormedRing
  • Matrix.frobeniusNormedAlgebra

• The $L^\infty$ operator norm:

  • Matrix.linftyOpSeminormedAddCommGroup
  • Matrix.linftyOpNormedAddCommGroup
  • Matrix.linftyOpNormedSpace
  • Matrix.linftyOpNonUnitalSemiNormedRing
  • Matrix.linftyOpSemiNormedRing
  • Matrix.linftyOpNonUnitalNormedRing
  • Matrix.linftyOpNormedRing
  • Matrix.linftyOpNormedAlgebra

These are not declared as instances because there are several natural choices for defining the norm of a matrix.

The elementwise supremum norm

def Matrix.seminormedAddCommGroup {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] : SeminormedAddCommGroup (Matrix m n α)

Seminormed group instance (using sup norm of sup norm) for matrices over a seminormed group. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

theorem Matrix.norm_def {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖A‖ = ‖fun i j => A i j‖

theorem Matrix.nnnorm_def {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖A‖₊ = ‖fun i j => A i j‖₊

theorem Matrix.norm_le_iff {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : ℝ} (hr : 0 ≤ r) {A : Matrix m n α} : ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r

theorem Matrix.nnnorm_le_iff {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : NNReal} {A : Matrix m n α} : ‖A‖₊ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ ≤ r

theorem Matrix.norm_lt_iff {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : ℝ} (hr : 0 < r) {A : Matrix m n α} : ‖A‖ < r ↔ ∀ (i : m) (j : n), ‖A i j‖ < r

theorem Matrix.nnnorm_lt_iff {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {r : NNReal} (hr : 0 < r) {A : Matrix m n α} : ‖A‖₊ < r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ < r

theorem Matrix.norm_entry_le_entrywise_sup_norm {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) {i : m} {j : n} : ‖A i j‖ ≤ ‖A‖

theorem Matrix.nnnorm_entry_le_entrywise_sup_nnnorm {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) {i : m} {j : n} : ‖A i j‖₊ ≤ ‖A‖₊

@[simp]

theorem Matrix.nnnorm_map_eq {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊) : ‖Matrix.map A f‖₊ = ‖A‖₊

@[simp]

theorem Matrix.norm_map_eq {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖ = ‖a‖) : ‖Matrix.map A f‖ = ‖A‖

@[simp]

theorem Matrix.nnnorm_transpose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖Matrix.transpose A‖₊ = ‖A‖₊

@[simp]

theorem Matrix.norm_transpose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖Matrix.transpose A‖ = ‖A‖

@[simp]

theorem Matrix.nnnorm_conjTranspose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) : ‖Matrix.conjTranspose A‖₊ = ‖A‖₊

@[simp]

theorem Matrix.norm_conjTranspose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) : ‖Matrix.conjTranspose A‖ = ‖A‖

instance Matrix.instNormedStarGroupMatrixSeminormedAddCommGroupInstStarAddMonoidMatrixAddMonoidToAddMonoidToSubNegMonoidToAddGroupToSeminormedAddGroup {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] : NormedStarGroup (Matrix m m α)

@[simp]

theorem Matrix.nnnorm_col {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] (v : m → α) : ‖Matrix.col v‖₊ = ‖v‖₊

@[simp]

theorem Matrix.norm_col {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] (v : m → α) : ‖Matrix.col v‖ = ‖v‖

@[simp]

theorem Matrix.nnnorm_row {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] (v : n → α) : ‖Matrix.row v‖₊ = ‖v‖₊

@[simp]

theorem Matrix.norm_row {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] (v : n → α) : ‖Matrix.row v‖ = ‖v‖

@[simp]

theorem Matrix.nnnorm_diagonal {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α) : ‖Matrix.diagonal v‖₊ = ‖v‖₊

@[simp]

theorem Matrix.norm_diagonal {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α) : ‖Matrix.diagonal v‖ = ‖v‖

instance Matrix.instNormOneClassMatrixToNormSeminormedAddCommGroupOneToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToAddCommGroup {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [Nonempty n] [DecidableEq n] [One α] [NormOneClass α] : NormOneClass (Matrix n n α)

Note this is safe as an instance as it carries no data.

def Matrix.normedAddCommGroup {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NormedAddCommGroup α] : NormedAddCommGroup (Matrix m n α)

Normed group instance (using sup norm of sup norm) for matrices over a normed group. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.normedSpace {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NormedField R] [SeminormedAddCommGroup α] [NormedSpace R α] : NormedSpace R (Matrix m n α)

Normed space instance (using sup norm of sup norm) for matrices over a normed space. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

The $L_\infty$ operator norm

This section defines the matrix norm $|A|_\infty = \operatorname{sup}i (\sum_j |A{ij}|)$.

Note that this is equivalent to the operator norm, considering $A$ as a linear map between two $L^\infty$ spaces.

def Matrix.linftyOpSeminormedAddCommGroup {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] : SeminormedAddCommGroup (Matrix m n α)

Seminormed group instance (using sup norm of L1 norm) for matrices over a seminormed group. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.linftyOpNormedAddCommGroup {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NormedAddCommGroup α] : NormedAddCommGroup (Matrix m n α)

Normed group instance (using sup norm of L1 norm) for matrices over a normed ring. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.linftyOpNormedSpace {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NormedField R] [SeminormedAddCommGroup α] [NormedSpace R α] : NormedSpace R (Matrix m n α)

Normed space instance (using sup norm of L1 norm) for matrices over a normed space. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

theorem Matrix.linfty_op_norm_def {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖A‖ = ↑(Finset.sup Finset.univ fun i => Finset.sum Finset.univ fun j => ‖A i j‖₊)

theorem Matrix.linfty_op_nnnorm_def {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖A‖₊ = Finset.sup Finset.univ fun i => Finset.sum Finset.univ fun j => ‖A i j‖₊

@[simp]

theorem Matrix.linfty_op_nnnorm_col {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] (v : m → α) : ‖Matrix.col v‖₊ = ‖v‖₊

@[simp]

theorem Matrix.linfty_op_norm_col {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] (v : m → α) : ‖Matrix.col v‖ = ‖v‖

@[simp]

theorem Matrix.linfty_op_nnnorm_row {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] (v : n → α) : ‖Matrix.row v‖₊ = Finset.sum Finset.univ fun i => ‖v i‖₊

@[simp]

theorem Matrix.linfty_op_norm_row {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] (v : n → α) : ‖Matrix.row v‖ = Finset.sum Finset.univ fun i => ‖v i‖

@[simp]

theorem Matrix.linfty_op_nnnorm_diagonal {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [DecidableEq m] (v : m → α) : ‖Matrix.diagonal v‖₊ = ‖v‖₊

@[simp]

theorem Matrix.linfty_op_norm_diagonal {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [DecidableEq m] (v : m → α) : ‖Matrix.diagonal v‖ = ‖v‖

theorem Matrix.linfty_op_nnnorm_mul {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [NonUnitalSeminormedRing α] (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊

theorem Matrix.linfty_op_norm_mul {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [NonUnitalSeminormedRing α] (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖ ≤ ‖A‖ * ‖B‖

theorem Matrix.linfty_op_nnnorm_mulVec {l : Type u_2} {m : Type u_3} {α : Type u_5} [Fintype l] [Fintype m] [NonUnitalSeminormedRing α] (A : Matrix l m α) (v : m → α) : ‖Matrix.mulVec A v‖₊ ≤ ‖A‖₊ * ‖v‖₊

theorem Matrix.linfty_op_norm_mulVec {l : Type u_2} {m : Type u_3} {α : Type u_5} [Fintype l] [Fintype m] [NonUnitalSeminormedRing α] (A : Matrix l m α) (v : m → α) : ‖Matrix.mulVec A v‖ ≤ ‖A‖ * ‖v‖

def Matrix.linftyOpNonUnitalSemiNormedRing {n : Type u_4} {α : Type u_5} [Fintype n] [NonUnitalSeminormedRing α] : NonUnitalSeminormedRing (Matrix n n α)

Seminormed non-unital ring instance (using sup norm of L1 norm) for matrices over a semi normed non-unital ring. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

instance Matrix.linfty_op_normOneClass {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedRing α] [NormOneClass α] [DecidableEq n] [Nonempty n] : NormOneClass (Matrix n n α)

The L₁-L∞ norm preserves one on non-empty matrices. Note this is safe as an instance, as it carries no data.

def Matrix.linftyOpSemiNormedRing {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedRing α] [DecidableEq n] : SeminormedRing (Matrix n n α)

Seminormed ring instance (using sup norm of L1 norm) for matrices over a semi normed ring. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.linftyOpNonUnitalNormedRing {n : Type u_4} {α : Type u_5} [Fintype n] [NonUnitalNormedRing α] : NonUnitalNormedRing (Matrix n n α)

Normed non-unital ring instance (using sup norm of L1 norm) for matrices over a normed non-unital ring. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.linftyOpNormedRing {n : Type u_4} {α : Type u_5} [Fintype n] [NormedRing α] [DecidableEq n] : NormedRing (Matrix n n α)

Normed ring instance (using sup norm of L1 norm) for matrices over a normed ring. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.linftyOpNormedAlgebra {R : Type u_1} {n : Type u_4} {α : Type u_5} [Fintype n] [NormedField R] [SeminormedRing α] [NormedAlgebra R α] [DecidableEq n] : NormedAlgebra R (Matrix n n α)

Normed algebra instance (using sup norm of L1 norm) for matrices over a normed algebra. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

The Frobenius norm

This is defined as $|A| = \sqrt{\sum_{i,j} |A_{ij}|^2}$. When the matrix is over the real or complex numbers, this norm is submultiplicative.

def Matrix.frobeniusSeminormedAddCommGroup {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] : SeminormedAddCommGroup (Matrix m n α)

Seminormed group instance (using frobenius norm) for matrices over a seminormed group. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.frobeniusNormedAddCommGroup {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NormedAddCommGroup α] : NormedAddCommGroup (Matrix m n α)

Normed group instance (using frobenius norm) for matrices over a normed group. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.frobeniusNormedSpace {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NormedField R] [SeminormedAddCommGroup α] [NormedSpace R α] : NormedSpace R (Matrix m n α)

Normed space instance (using frobenius norm) for matrices over a normed space. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

theorem Matrix.frobenius_nnnorm_def {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖A‖₊ = (Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => ‖A i j‖₊ ^ 2) ^ (1 / 2)

theorem Matrix.frobenius_norm_def {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖A‖ = (Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => ‖A i j‖ ^ 2) ^ (1 / 2)

@[simp]

theorem Matrix.frobenius_nnnorm_map_eq {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊) : ‖Matrix.map A f‖₊ = ‖A‖₊

@[simp]

theorem Matrix.frobenius_norm_map_eq {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖ = ‖a‖) : ‖Matrix.map A f‖ = ‖A‖

@[simp]

theorem Matrix.frobenius_nnnorm_transpose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖Matrix.transpose A‖₊ = ‖A‖₊

@[simp]

theorem Matrix.frobenius_norm_transpose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α) : ‖Matrix.transpose A‖ = ‖A‖

@[simp]

theorem Matrix.frobenius_nnnorm_conjTranspose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) : ‖Matrix.conjTranspose A‖₊ = ‖A‖₊

@[simp]

theorem Matrix.frobenius_norm_conjTranspose {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) : ‖Matrix.conjTranspose A‖ = ‖A‖

instance Matrix.frobenius_normedStarGroup {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] : NormedStarGroup (Matrix m m α)

@[simp]

theorem Matrix.frobenius_norm_row {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] (v : m → α) : ‖Matrix.row v‖ = ‖↑(WithLp.equiv 2 (m → α)).symm v‖

@[simp]

theorem Matrix.frobenius_nnnorm_row {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] (v : m → α) : ‖Matrix.row v‖₊ = ‖↑(WithLp.equiv 2 (m → α)).symm v‖₊

@[simp]

theorem Matrix.frobenius_norm_col {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] (v : n → α) : ‖Matrix.col v‖ = ‖↑(WithLp.equiv 2 (n → α)).symm v‖

@[simp]

theorem Matrix.frobenius_nnnorm_col {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] (v : n → α) : ‖Matrix.col v‖₊ = ‖↑(WithLp.equiv 2 (n → α)).symm v‖₊

@[simp]

theorem Matrix.frobenius_nnnorm_diagonal {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α) : ‖Matrix.diagonal v‖₊ = ‖↑(WithLp.equiv 2 (n → α)).symm v‖₊

@[simp]

theorem Matrix.frobenius_norm_diagonal {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α) : ‖Matrix.diagonal v‖ = ‖↑(WithLp.equiv 2 (n → α)).symm v‖

theorem Matrix.frobenius_nnnorm_one {n : Type u_4} {α : Type u_5} [Fintype n] [DecidableEq n] [SeminormedAddCommGroup α] [One α] : ‖1‖₊ = ↑NNReal.sqrt ↑(Fintype.card n) * ‖1‖₊

theorem Matrix.frobenius_nnnorm_mul {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [IsROrC α] (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊

theorem Matrix.frobenius_norm_mul {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [IsROrC α] (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖ ≤ ‖A‖ * ‖B‖

def Matrix.frobeniusNormedRing {m : Type u_3} {α : Type u_5} [Fintype m] [IsROrC α] [DecidableEq m] : NormedRing (Matrix m m α)

Normed ring instance (using frobenius norm) for matrices over ℝ or ℂ. Not declared as an instance because there are several natural choices for defining the norm of a matrix.

def Matrix.frobeniusNormedAlgebra {R : Type u_1} {m : Type u_3} {α : Type u_5} [Fintype m] [IsROrC α] [DecidableEq m] [NormedField R] [NormedAlgebra R α] : NormedAlgebra R (Matrix m m α)

Normed algebra instance (using frobenius norm) for matrices over ℝ or ℂ. Not declared as an instance because there are several natural choices for defining the norm of a matrix.