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 #

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
    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
    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

    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‖₊
    theorem Matrix.norm_le_iff {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] {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 : } (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} :
    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} :
    @[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 : αβ) (hf : ∀ (a : α), f a‖₊ = 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 : αβ) (hf : ∀ (a : α), f a = a) :
    A.map 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) :
    A.transpose‖₊ = 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) :
    A.transpose = 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) :
    A.conjTranspose‖₊ = 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) :
    A.conjTranspose = A
    Equations
    • =
    @[simp]
    theorem Matrix.nnnorm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : mα) :
    @[simp]
    theorem Matrix.norm_col {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : mα) :
    @[simp]
    theorem Matrix.nnnorm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : nα) :
    @[simp]
    theorem Matrix.norm_row {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : nα) :
    @[simp]
    @[simp]
    theorem Matrix.norm_diagonal {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : nα) :

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

    Equations
    • =
    def Matrix.normedAddCommGroup {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NormedAddCommGroup 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
      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.

      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.

        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

          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
            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.

            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
              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‖₊)
              @[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.

              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‖₊
              @[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.

              @[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 : mα) :
              @[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 : mα) :

              Alias of Matrix.linfty_opNNNorm_col.

              @[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 : mα) :
              @[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 : mα) :

              Alias of Matrix.linfty_opNorm_col.

              @[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 : nα) :
              Matrix.row ι v‖₊ = i : n, v i‖₊
              @[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 : nα) :
              Matrix.row ι v‖₊ = i : n, v i‖₊

              Alias of Matrix.linfty_opNNNorm_row.

              @[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 : nα) :
              Matrix.row ι v = i : n, v i
              @[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 : nα) :
              Matrix.row ι v = i : n, v i

              Alias of Matrix.linfty_opNorm_row.

              @[deprecated Matrix.linfty_opNNNorm_diagonal]

              Alias of Matrix.linfty_opNNNorm_diagonal.

              @[simp]
              @[deprecated Matrix.linfty_opNorm_diagonal]

              Alias of Matrix.linfty_opNorm_diagonal.

              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) :
              @[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) :

              Alias of Matrix.linfty_opNNNorm_mul.

              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) :
              @[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) :

              Alias of Matrix.linfty_opNorm_mul.

              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 : mα) :
              @[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 : mα) :

              Alias of Matrix.linfty_opNNNorm_mulVec.

              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 : mα) :
              A.mulVec v A * v
              @[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 : mα) :
              A.mulVec v A * v

              Alias of Matrix.linfty_opNorm_mulVec.

              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
              Instances For
                instance Matrix.linfty_opNormOneClass {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedRing α] [NormOneClass α] [DecidableEq n] [Nonempty n] :

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

                Equations
                • =

                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
                Instances For

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

                    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
                    Instances For
                      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] :

                      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
                      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).

                        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 α] (A : Matrix m n α) :
                        A‖₊ = { toLinearMap := A.mulVecLin, cont := }‖₊
                        @[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 α] (A : Matrix m n α) :
                        A‖₊ = { toLinearMap := A.mulVecLin, cont := }‖₊

                        Alias of Matrix.linfty_opNNNorm_eq_opNNNorm.

                        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 α] (A : Matrix m n α) :
                        A = { toLinearMap := A.mulVecLin, cont := }
                        @[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 α] (A : Matrix m n α) :
                        A = { toLinearMap := A.mulVecLin, cont := }

                        Alias of Matrix.linfty_opNorm_eq_opNorm.

                        @[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 α] [DecidableEq n] (f : (nα) →L[α] mα) :
                        LinearMap.toMatrix' f‖₊ = f‖₊
                        @[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 α] [DecidableEq n] (f : (nα) →L[α] mα) :
                        LinearMap.toMatrix' f‖₊ = f‖₊

                        Alias of Matrix.linfty_opNNNorm_toMatrix.

                        @[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 α] [DecidableEq n] (f : (nα) →L[α] mα) :
                        LinearMap.toMatrix' f = f
                        @[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 α] [DecidableEq n] (f : (nα) →L[α] mα) :
                        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.

                        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

                          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
                            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.

                            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
                              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)
                              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)
                              @[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 : αβ) (hf : ∀ (a : α), f a‖₊ = 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 : αβ) (hf : ∀ (a : α), f a = a) :
                              A.map 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) :
                              A.transpose‖₊ = 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) :
                              A.transpose = 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) :
                              A.conjTranspose‖₊ = 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) :
                              A.conjTranspose = A
                              Equations
                              • =
                              @[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 : mα) :
                              Matrix.row ι v = (WithLp.equiv 2 (mα)).symm v
                              @[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 : mα) :
                              Matrix.row ι v‖₊ = (WithLp.equiv 2 (mα)).symm v‖₊
                              @[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 : nα) :
                              Matrix.col ι v = (WithLp.equiv 2 (nα)).symm v
                              @[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 : nα) :
                              Matrix.col ι v‖₊ = (WithLp.equiv 2 (nα)).symm v‖₊
                              @[simp]
                              theorem Matrix.frobenius_nnnorm_diagonal {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : nα) :
                              @[simp]
                              theorem Matrix.frobenius_norm_diagonal {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : nα) :
                              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] [RCLike α] (A : Matrix l m α) (B : Matrix m n α) :
                              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) :
                              def Matrix.frobeniusNormedRing {m : Type u_3} {α : Type u_5} [Fintype m] [RCLike α] [DecidableEq m] :

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

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                                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] :

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

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