Documentation

Mathlib.Analysis.Matrix

Matrices as a normed space #

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

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

The norm induced by the identification of Matrix m n 𝕜 with EuclideanSpace n 𝕜 →L[𝕜] EuclideanSpace m 𝕜 (i.e., the ℓ² operator norm) can be found in Analysis.CstarAlgebra.Matrix. It is separated to avoid extraneous imports in this file.

The elementwise supremum norm #

source
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.

Equations
  • Matrix.seminormedAddCommGroup = Pi.seminormedAddCommGroup
Instances For
    source
    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 : m) (j : n) => A i j‖
    source
    theorem Matrix.norm_eq_sup_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) :
    ‖A‖ = ↑(Finset.univ.sup fun (i : m) => Finset.univ.sup fun (j : n) => ‖A i j‖₊)

    The norm of a matrix is the sup of the sup of the nnnorm of the entries

    source
    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 : m) (j : n) => A i j‖₊
    source
    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
    source
    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
    source
    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
    source
    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
    source
    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‖
    source
    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‖₊
    source
    @[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‖₊) :
    ‖A.map f‖₊ = ‖A‖₊
    source
    @[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‖) :
    ‖A.map f‖ = ‖A‖
    source
    @[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) :
    ‖A.transpose‖₊ = ‖A‖₊
    source
    @[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) :
    ‖A.transpose‖ = ‖A‖
    source
    @[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) :
    ‖A.conjTranspose‖₊ = ‖A‖₊
    source
    @[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) :
    ‖A.conjTranspose‖ = ‖A‖
    source
    instance Matrix.instNormedStarGroup {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] :
    NormedStarGroup (Matrix m m α)
    Equations
    • ⋯ = ⋯
    source
    @[simp]
    theorem Matrix.nnnorm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
    ‖Matrix.col ι v‖₊ = ‖v‖₊
    source
    @[simp]
    theorem Matrix.norm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
    ‖Matrix.col ι v‖ = ‖v‖
    source
    @[simp]
    theorem Matrix.nnnorm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
    ‖Matrix.row ι v‖₊ = ‖v‖₊
    source
    @[simp]
    theorem Matrix.norm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
    ‖Matrix.row ι v‖ = ‖v‖
    source
    @[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‖₊
    source
    @[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‖
    source
    instance Matrix.instNormOneClassOfNonempty {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.

    Equations
    • ⋯ = ⋯
    source
    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.

    Equations
    • Matrix.normedAddCommGroup = Pi.normedAddCommGroup
    Instances For
      source
      theorem Matrix.boundedSMul {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α] [BoundedSMul R α] :
      BoundedSMul R (Matrix m n α)

      This applies to the sup norm of sup norm.

      source
      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.

      Equations
      • Matrix.normedSpace = Pi.normedSpace
      Instances For

        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.

        source
        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.

        Equations
        • Matrix.linftyOpSeminormedAddCommGroup = inferInstance
        Instances For
          source
          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.

          Equations
          • Matrix.linftyOpNormedAddCommGroup = inferInstance
          Instances For
            source
            theorem Matrix.linftyOpBoundedSMul {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α] [BoundedSMul R α] :
            BoundedSMul R (Matrix m n α)

            This applies to the sup norm of L1 norm.

            source
            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.

            Equations
            • Matrix.linftyOpNormedSpace = inferInstance
            Instances For
              source
              theorem Matrix.linfty_opNorm_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.univ.sup fun (i : m) => ∑ j : n, ‖A i j‖₊)
              source
              @[deprecated Matrix.linfty_opNorm_def]
              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.univ.sup fun (i : m) => ∑ j : n, ‖A i j‖₊)

              Alias of Matrix.linfty_opNorm_def.

              source
              theorem Matrix.linfty_opNNNorm_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.univ.sup fun (i : m) => ∑ j : n, ‖A i j‖₊
              source
              @[deprecated Matrix.linfty_opNNNorm_def]
              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.univ.sup fun (i : m) => ∑ j : n, ‖A i j‖₊

              Alias of Matrix.linfty_opNNNorm_def.

              source
              @[simp]
              theorem Matrix.linfty_opNNNorm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
              ‖Matrix.col ι v‖₊ = ‖v‖₊
              source
              @[deprecated Matrix.linfty_opNNNorm_col]
              theorem Matrix.linfty_op_nnnorm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
              ‖Matrix.col ι v‖₊ = ‖v‖₊

              Alias of Matrix.linfty_opNNNorm_col.

              source
              @[simp]
              theorem Matrix.linfty_opNorm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
              ‖Matrix.col ι v‖ = ‖v‖
              source
              @[deprecated Matrix.linfty_opNorm_col]
              theorem Matrix.linfty_op_norm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
              ‖Matrix.col ι v‖ = ‖v‖

              Alias of Matrix.linfty_opNorm_col.

              source
              @[simp]
              theorem Matrix.linfty_opNNNorm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
              ‖Matrix.row ι v‖₊ = ∑ i : n, ‖v i‖₊
              source
              @[deprecated Matrix.linfty_opNNNorm_row]
              theorem Matrix.linfty_op_nnnorm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
              ‖Matrix.row ι v‖₊ = ∑ i : n, ‖v i‖₊

              Alias of Matrix.linfty_opNNNorm_row.

              source
              @[simp]
              theorem Matrix.linfty_opNorm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
              ‖Matrix.row ι v‖ = ∑ i : n, ‖v i‖
              source
              @[deprecated Matrix.linfty_opNorm_row]
              theorem Matrix.linfty_op_norm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
              ‖Matrix.row ι v‖ = ∑ i : n, ‖v i‖

              Alias of Matrix.linfty_opNorm_row.

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

              Alias of Matrix.linfty_opNNNorm_diagonal.

              source
              @[simp]
              theorem Matrix.linfty_opNorm_diagonal {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [DecidableEq m] (v : m → α) :
              ‖Matrix.diagonal v‖ = ‖v‖
              source
              @[deprecated Matrix.linfty_opNorm_diagonal]
              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‖

              Alias of Matrix.linfty_opNorm_diagonal.

              source
              theorem Matrix.linfty_opNNNorm_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‖₊
              source
              @[deprecated Matrix.linfty_opNNNorm_mul]
              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‖₊

              Alias of Matrix.linfty_opNNNorm_mul.

              source
              theorem Matrix.linfty_opNorm_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‖
              source
              @[deprecated Matrix.linfty_opNorm_mul]
              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‖

              Alias of Matrix.linfty_opNorm_mul.

              source
              theorem Matrix.linfty_opNNNorm_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) :
              ‖A.mulVec v‖₊ ≤ ‖A‖₊ * ‖v‖₊
              source
              @[deprecated Matrix.linfty_opNNNorm_mulVec]
              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) :
              ‖A.mulVec v‖₊ ≤ ‖A‖₊ * ‖v‖₊

              Alias of Matrix.linfty_opNNNorm_mulVec.

              source
              theorem Matrix.linfty_opNorm_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) :
              ‖A.mulVec v‖ ≤ ‖A‖ * ‖v‖
              source
              @[deprecated Matrix.linfty_opNorm_mulVec]
              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) :
              ‖A.mulVec v‖ ≤ ‖A‖ * ‖v‖

              Alias of Matrix.linfty_opNorm_mulVec.

              source
              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.

              Equations
              • Matrix.linftyOpNonUnitalSemiNormedRing = let __src := Matrix.linftyOpSeminormedAddCommGroup; let __src_1 := Matrix.instNonUnitalRing; NonUnitalSeminormedRing.mk ⋯ ⋯
              Instances For
                source
                instance Matrix.linfty_opNormOneClass {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.

                Equations
                • ⋯ = ⋯
                source
                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.

                Equations
                • Matrix.linftyOpSemiNormedRing = let __src := Matrix.linftyOpNonUnitalSemiNormedRing; let __src_1 := Matrix.instRing; SeminormedRing.mk ⋯ ⋯
                Instances For
                  source
                  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.

                  Equations
                  • Matrix.linftyOpNonUnitalNormedRing = let __src := Matrix.linftyOpNonUnitalSemiNormedRing; NonUnitalNormedRing.mk ⋯ ⋯
                  Instances For
                    source
                    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.

                    Equations
                    • Matrix.linftyOpNormedRing = let __src := Matrix.linftyOpSemiNormedRing; NormedRing.mk ⋯ ⋯
                    Instances For
                      source
                      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.

                      Equations
                      • Matrix.linftyOpNormedAlgebra = let __src := Matrix.linftyOpNormedSpace; let __src_1 := Matrix.instAlgebra; NormedAlgebra.mk ⋯
                      Instances For

                        For a matrix over a field, the norm defined in this section agrees with the operator norm on ContinuousLinearMaps between function types (which have the infinity norm).

                        source
                        theorem Matrix.linfty_opNNNorm_eq_opNNNorm {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] (A : Matrix m n α) :
                        ‖A‖₊ = ‖{ toLinearMap := A.mulVecLin, cont := ⋯ }‖₊
                        source
                        @[deprecated Matrix.linfty_opNNNorm_eq_opNNNorm]
                        theorem Matrix.linfty_op_nnnorm_eq_op_nnnorm {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] (A : Matrix m n α) :
                        ‖A‖₊ = ‖{ toLinearMap := A.mulVecLin, cont := ⋯ }‖₊

                        Alias of Matrix.linfty_opNNNorm_eq_opNNNorm.

                        source
                        theorem Matrix.linfty_opNorm_eq_opNorm {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] (A : Matrix m n α) :
                        ‖A‖ = ‖{ toLinearMap := A.mulVecLin, cont := ⋯ }‖
                        source
                        @[deprecated Matrix.linfty_opNorm_eq_opNorm]
                        theorem Matrix.linfty_op_norm_eq_op_norm {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] (A : Matrix m n α) :
                        ‖A‖ = ‖{ toLinearMap := A.mulVecLin, cont := ⋯ }‖

                        Alias of Matrix.linfty_opNorm_eq_opNorm.

                        source
                        @[simp]
                        theorem Matrix.linfty_opNNNorm_toMatrix {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] [DecidableEq n] (f : (n → α) →L[α] m → α) :
                        ‖LinearMap.toMatrix' ↑f‖₊ = ‖f‖₊
                        source
                        @[deprecated Matrix.linfty_opNNNorm_toMatrix]
                        theorem Matrix.linfty_op_nnnorm_toMatrix {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] [DecidableEq n] (f : (n → α) →L[α] m → α) :
                        ‖LinearMap.toMatrix' ↑f‖₊ = ‖f‖₊

                        Alias of Matrix.linfty_opNNNorm_toMatrix.

                        source
                        @[simp]
                        theorem Matrix.linfty_opNorm_toMatrix {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] [DecidableEq n] (f : (n → α) →L[α] m → α) :
                        ‖LinearMap.toMatrix' ↑f‖ = ‖f‖
                        source
                        @[deprecated Matrix.linfty_opNorm_toMatrix]
                        theorem Matrix.linfty_op_norm_toMatrix {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] [DecidableEq n] (f : (n → α) →L[α] m → α) :
                        ‖LinearMap.toMatrix' ↑f‖ = ‖f‖

                        Alias of Matrix.linfty_opNorm_toMatrix.

                        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.

                        source
                        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.

                        Equations
                        Instances For
                          source
                          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.

                          Equations
                          • Matrix.frobeniusNormedAddCommGroup = inferInstance
                          Instances For
                            source
                            theorem Matrix.frobeniusBoundedSMul {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α] [BoundedSMul R α] :
                            BoundedSMul R (Matrix m n α)

                            This applies to the frobenius norm.

                            source
                            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.

                            Equations
                            • Matrix.frobeniusNormedSpace = inferInstance
                            Instances For
                              source
                              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‖₊ = (∑ i : m, ∑ j : n, ‖A i j‖₊ ^ 2) ^ (1 / 2)
                              source
                              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‖ = (∑ i : m, ∑ j : n, ‖A i j‖ ^ 2) ^ (1 / 2)
                              source
                              @[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‖₊) :
                              ‖A.map f‖₊ = ‖A‖₊
                              source
                              @[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‖) :
                              ‖A.map f‖ = ‖A‖
                              source
                              @[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) :
                              ‖A.transpose‖₊ = ‖A‖₊
                              source
                              @[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) :
                              ‖A.transpose‖ = ‖A‖
                              source
                              @[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) :
                              ‖A.conjTranspose‖₊ = ‖A‖₊
                              source
                              @[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) :
                              ‖A.conjTranspose‖ = ‖A‖
                              source
                              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)
                              Equations
                              • ⋯ = ⋯
                              source
                              @[simp]
                              theorem Matrix.frobenius_norm_row {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
                              ‖Matrix.row ι v‖ = ‖(WithLp.equiv 2 (m → α)).symm v‖
                              source
                              @[simp]
                              theorem Matrix.frobenius_nnnorm_row {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α) :
                              ‖Matrix.row ι v‖₊ = ‖(WithLp.equiv 2 (m → α)).symm v‖₊
                              source
                              @[simp]
                              theorem Matrix.frobenius_norm_col {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
                              ‖Matrix.col ι v‖ = ‖(WithLp.equiv 2 (n → α)).symm v‖
                              source
                              @[simp]
                              theorem Matrix.frobenius_nnnorm_col {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α) :
                              ‖Matrix.col ι v‖₊ = ‖(WithLp.equiv 2 (n → α)).symm v‖₊
                              source
                              @[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‖₊
                              source
                              @[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‖
                              source
                              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‖₊
                              source
                              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] [RCLike α] (A : Matrix l m α) (B : Matrix m n α) :
                              ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊
                              source
                              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] [RCLike α] (A : Matrix l m α) (B : Matrix m n α) :
                              ‖A * B‖ ≤ ‖A‖ * ‖B‖
                              source
                              def Matrix.frobeniusNormedRing {m : Type u_3} {α : Type u_5} [Fintype m] [RCLike 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.

                              Equations
                              • Matrix.frobeniusNormedRing = let __src := Matrix.frobeniusSeminormedAddCommGroup; let __src_1 := Matrix.instRing; NormedRing.mk ⋯ ⋯
                              Instances For
                                source
                                def Matrix.frobeniusNormedAlgebra {R : Type u_1} {m : Type u_3} {α : Type u_5} [Fintype m] [RCLike 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.

                                Equations
                                • Matrix.frobeniusNormedAlgebra = let __src := Matrix.frobeniusNormedSpace; let __src_1 := Matrix.instAlgebra; NormedAlgebra.mk ⋯
                                Instances For