Documentation

Mathlib.LinearAlgebra.Matrix.Trace

Trace of a matrix #

This file defines the trace of a matrix, the map sending a matrix to the sum of its diagonal entries.

See also LinearAlgebra.Trace for the trace of an endomorphism.

Tags #

matrix, trace, diagonal

def Matrix.trace {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommMonoid R] (A : Matrix n n R) :
R

The trace of a square matrix. For more bundled versions, see:

Equations
Instances For
    theorem Matrix.trace_diagonal {R : Type u_6} [AddCommMonoid R] {o : Type u_8} [Fintype o] [DecidableEq o] (d : oR) :
    Matrix.trace (Matrix.diagonal d) = Finset.sum Finset.univ fun (i : o) => d i
    @[simp]
    theorem Matrix.trace_zero (n : Type u_3) (R : Type u_6) [Fintype n] [AddCommMonoid R] :
    @[simp]
    theorem Matrix.trace_eq_zero_of_isEmpty {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommMonoid R] [IsEmpty n] (A : Matrix n n R) :
    @[simp]
    theorem Matrix.trace_add {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommMonoid R] (A : Matrix n n R) (B : Matrix n n R) :
    @[simp]
    theorem Matrix.trace_smul {n : Type u_3} {α : Type u_5} {R : Type u_6} [Fintype n] [AddCommMonoid R] [Monoid α] [DistribMulAction α R] (r : α) (A : Matrix n n R) :
    @[simp]
    @[simp]
    def Matrix.traceAddMonoidHom (n : Type u_3) (R : Type u_6) [Fintype n] [AddCommMonoid R] :
    Matrix n n R →+ R

    Matrix.trace as an AddMonoidHom

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      @[simp]
      theorem Matrix.traceLinearMap_apply (n : Type u_3) (α : Type u_5) (R : Type u_6) [Fintype n] [AddCommMonoid R] [Semiring α] [Module α R] (A : Matrix n n R) :
      def Matrix.traceLinearMap (n : Type u_3) (α : Type u_5) (R : Type u_6) [Fintype n] [AddCommMonoid R] [Semiring α] [Module α R] :
      Matrix n n R →ₗ[α] R

      Matrix.trace as a LinearMap

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        @[simp]
        theorem Matrix.trace_list_sum {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommMonoid R] (l : List (Matrix n n R)) :
        Matrix.trace (List.sum l) = List.sum (List.map Matrix.trace l)
        @[simp]
        theorem Matrix.trace_multiset_sum {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommMonoid R] (s : Multiset (Matrix n n R)) :
        @[simp]
        theorem Matrix.trace_sum {ι : Type u_1} {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommMonoid R] (s : Finset ι) (f : ιMatrix n n R) :
        Matrix.trace (Finset.sum s fun (i : ι) => f i) = Finset.sum s fun (i : ι) => Matrix.trace (f i)
        theorem AddMonoidHom.map_trace {n : Type u_3} {R : Type u_6} {S : Type u_7} [Fintype n] [AddCommMonoid R] [AddCommMonoid S] (f : R →+ S) (A : Matrix n n R) :
        theorem Matrix.trace_blockDiagonal {n : Type u_3} {p : Type u_4} {R : Type u_6} [Fintype n] [Fintype p] [AddCommMonoid R] [DecidableEq p] (M : pMatrix n n R) :
        Matrix.trace (Matrix.blockDiagonal M) = Finset.sum Finset.univ fun (i : p) => Matrix.trace (M i)
        theorem Matrix.trace_blockDiagonal' {p : Type u_4} {R : Type u_6} [Fintype p] [AddCommMonoid R] [DecidableEq p] {m : pType u_8} [(i : p) → Fintype (m i)] (M : (i : p) → Matrix (m i) (m i) R) :
        Matrix.trace (Matrix.blockDiagonal' M) = Finset.sum Finset.univ fun (i : p) => Matrix.trace (M i)
        @[simp]
        theorem Matrix.trace_sub {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommGroup R] (A : Matrix n n R) (B : Matrix n n R) :
        @[simp]
        theorem Matrix.trace_neg {n : Type u_3} {R : Type u_6} [Fintype n] [AddCommGroup R] (A : Matrix n n R) :
        @[simp]
        theorem Matrix.trace_one {n : Type u_3} {R : Type u_6} [Fintype n] [DecidableEq n] [AddCommMonoidWithOne R] :
        @[simp]
        theorem Matrix.trace_transpose_mul {m : Type u_2} {n : Type u_3} {R : Type u_6} [Fintype m] [Fintype n] [AddCommMonoid R] [Mul R] (A : Matrix m n R) (B : Matrix n m R) :
        theorem Matrix.trace_mul_comm {m : Type u_2} {n : Type u_3} {R : Type u_6} [Fintype m] [Fintype n] [AddCommMonoid R] [CommSemigroup R] (A : Matrix m n R) (B : Matrix n m R) :
        theorem Matrix.trace_mul_cycle {m : Type u_2} {n : Type u_3} {p : Type u_4} {R : Type u_6} [Fintype m] [Fintype n] [Fintype p] [NonUnitalCommSemiring R] (A : Matrix m n R) (B : Matrix n p R) (C : Matrix p m R) :
        Matrix.trace (A * B * C) = Matrix.trace (C * A * B)
        theorem Matrix.trace_mul_cycle' {m : Type u_2} {n : Type u_3} {p : Type u_4} {R : Type u_6} [Fintype m] [Fintype n] [Fintype p] [NonUnitalCommSemiring R] (A : Matrix m n R) (B : Matrix n p R) (C : Matrix p m R) :
        Matrix.trace (A * (B * C)) = Matrix.trace (C * (A * B))
        @[simp]
        theorem Matrix.trace_col_mul_row {n : Type u_3} {R : Type u_6} [Fintype n] [NonUnitalNonAssocSemiring R] (a : nR) (b : nR) :
        theorem Matrix.trace_submatrix_succ {R : Type u_6} {n : } [NonUnitalNonAssocSemiring R] (M : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R) :
        M 0 0 + Matrix.trace (Matrix.submatrix M Fin.succ Fin.succ) = Matrix.trace M

        Special cases for Fin n #

        While simp [Fin.sum_univ_succ] can prove these, we include them for convenience and consistency with Matrix.det_fin_two etc.

        theorem Matrix.trace_fin_zero {R : Type u_6} [AddCommMonoid R] (A : Matrix (Fin 0) (Fin 0) R) :
        theorem Matrix.trace_fin_one {R : Type u_6} [AddCommMonoid R] (A : Matrix (Fin 1) (Fin 1) R) :
        Matrix.trace A = A 0 0
        theorem Matrix.trace_fin_two {R : Type u_6} [AddCommMonoid R] (A : Matrix (Fin 2) (Fin 2) R) :
        Matrix.trace A = A 0 0 + A 1 1
        theorem Matrix.trace_fin_three {R : Type u_6} [AddCommMonoid R] (A : Matrix (Fin 3) (Fin 3) R) :
        Matrix.trace A = A 0 0 + A 1 1 + A 2 2