Linear maps and matrices
This file defines the maps to send matrices to a linear map, and to send linear maps between modules with a finite bases to matrices. This defines a linear equivalence between linear maps between finite-dimensional vector spaces and matrices indexed by the respective bases.
It also defines the trace of an endomorphism, and the determinant of a family of vectors with respect to some basis.
Some results are proved about the linear map corresponding to a
diagonal matrix (range, ker and rank).
Main definitions
In the list below, and in all this file, R is a commutative ring (semiring
is sometimes enough), M and its variations are R-modules, ι, κ, n and m are finite
types used for indexing.
linear_map.to_matrix: given basesv₁ : ι → M₁andv₂ : κ → M₂, theR-linear equivalence fromM₁ →ₗ[R] M₂tomatrix κ ι Rmatrix.to_lin: the inverse oflinear_map.to_matrixlinear_map.to_matrix': theR-linear equivalence from(n → R) →ₗ[R] (m → R)tomatrix n m R(with the standard basis onn → Randm → R)matrix.to_lin': the inverse oflinear_map.to_matrix'alg_equiv_matrix: given a basis indexed byn, theR-algebra equivalence betweenR-endomorphisms ofMandmatrix n n Rmatrix.trace: the trace of a square matrixlinear_map.trace: the trace of an endomorphismis_basis.to_matrix: the matrix whose columns are a given family of vectors in a given basisis_basis.to_matrix_equiv: given a basis, the linear equivalence between families of vectors and matrices arising fromis_basis.to_matrixis_basis.det: the determinant of a family of vectors with respect to a basis, as a multilinear map
Tags
linear_map, matrix, linear_equiv, diagonal, det, trace
@[instance]
def
matrix.fintype
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq m]
[decidable_eq n]
(R : Type u_1)
[fintype R] :
Equations
- matrix.fintype R = _.mpr pi.fintype
@[simp]
theorem
matrix.mul_vec_lin_apply
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
(M : matrix m n R)
(v : n → R) :
⇑(M.mul_vec_lin) v = M.mul_vec v
@[simp]
theorem
matrix.mul_vec_std_basis
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
(M : matrix m n R)
(i : m)
(j : n) :
M.mul_vec (⇑(linear_map.std_basis R (λ (_x : n), R) j) 1) i = M i j
def
linear_map.to_matrix'
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n] :
Linear maps (n → R) →ₗ[R] (m → R) are linearly equivalent to matrix m n R.
Equations
- linear_map.to_matrix' = {to_fun := λ (f : (n → R) →ₗ[R] m → R) (i : m) (j : n), ⇑f (⇑(linear_map.std_basis R (λ (_x : n), R) j) 1) i, map_add' := _, map_smul' := _, inv_fun := matrix.mul_vec_lin _inst_4, left_inv := _, right_inv := _}
def
matrix.to_lin'
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n] :
A matrix m n R is linearly equivalent to a linear map (n → R) →ₗ[R] (m → R).
Equations
@[simp]
theorem
linear_map.to_matrix'_symm
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n] :
@[simp]
theorem
matrix.to_lin'_symm
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n] :
@[simp]
theorem
linear_map.to_matrix'_to_lin'
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
(M : matrix m n R) :
@[simp]
theorem
matrix.to_lin'_to_matrix'
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
(f : (n → R) →ₗ[R] m → R) :
@[simp]
theorem
linear_map.to_matrix'_apply
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
(f : (n → R) →ₗ[R] m → R)
(i : m)
(j : n) :
@[simp]
theorem
matrix.to_lin'_apply
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
(M : matrix m n R)
(v : n → R) :
⇑(⇑matrix.to_lin' M) v = M.mul_vec v
@[simp]
theorem
matrix.to_lin'_one
{R : Type u_1}
[comm_ring R]
{n : Type u_4}
[fintype n]
[decidable_eq n] :
@[simp]
theorem
linear_map.to_matrix'_id
{R : Type u_1}
[comm_ring R]
{n : Type u_4}
[fintype n]
[decidable_eq n] :
@[simp]
theorem
matrix.to_lin'_mul
{R : Type u_1}
[comm_ring R]
{l : Type u_2}
{m : Type u_3}
{n : Type u_4}
[fintype l]
[fintype m]
[fintype n]
[decidable_eq n]
[decidable_eq m]
(M : matrix l m R)
(N : matrix m n R) :
⇑matrix.to_lin' (M ⬝ N) = (⇑matrix.to_lin' M).comp (⇑matrix.to_lin' N)
theorem
linear_map.to_matrix'_comp
{R : Type u_1}
[comm_ring R]
{l : Type u_2}
{m : Type u_3}
{n : Type u_4}
[fintype l]
[fintype m]
[fintype n]
[decidable_eq n]
[decidable_eq l]
(f : (n → R) →ₗ[R] m → R)
(g : (l → R) →ₗ[R] n → R) :
theorem
linear_map.to_matrix'_mul
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
[fintype m]
[decidable_eq m]
(f g : (m → R) →ₗ[R] m → R) :
def
linear_map.to_matrix
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂) :
Given bases of two modules M₁ and M₂ over a commutative ring R, we get a linear
equivalence between linear maps M₁ →ₗ M₂ and matrices over R indexed by the bases.
Equations
- linear_map.to_matrix hv₁ hv₂ = (hv₁.equiv_fun.arrow_congr hv₂.equiv_fun).trans linear_map.to_matrix'
def
matrix.to_lin
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂) :
Given bases of two modules M₁ and M₂ over a commutative ring R, we get a linear
equivalence between matrices over R indexed by the bases and linear maps M₁ →ₗ M₂.
Equations
- matrix.to_lin hv₁ hv₂ = (linear_map.to_matrix hv₁ hv₂).symm
@[simp]
theorem
linear_map.to_matrix_symm
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂) :
(linear_map.to_matrix hv₁ hv₂).symm = matrix.to_lin hv₁ hv₂
@[simp]
theorem
matrix.to_lin_symm
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂) :
(matrix.to_lin hv₁ hv₂).symm = linear_map.to_matrix hv₁ hv₂
@[simp]
theorem
matrix.to_lin_to_matrix
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(f : M₁ →ₗ[R] M₂) :
⇑(matrix.to_lin hv₁ hv₂) (⇑(linear_map.to_matrix hv₁ hv₂) f) = f
@[simp]
theorem
linear_map.to_matrix_to_lin
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(M : matrix m n R) :
⇑(linear_map.to_matrix hv₁ hv₂) (⇑(matrix.to_lin hv₁ hv₂) M) = M
theorem
linear_map.to_matrix_apply
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(f : M₁ →ₗ[R] M₂)
(i : m)
(j : n) :
theorem
linear_map.to_matrix_transpose_apply
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(f : M₁ →ₗ[R] M₂)
(j : n) :
theorem
linear_map.to_matrix_apply'
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(f : M₁ →ₗ[R] M₂)
(i : m)
(j : n) :
theorem
linear_map.to_matrix_transpose_apply'
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(f : M₁ →ₗ[R] M₂)
(j : n) :
theorem
matrix.to_lin_apply
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(M : matrix m n R)
(v : M₁) :
@[simp]
theorem
matrix.to_lin_self
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
(M : matrix m n R)
(i : n) :
⇑(⇑(matrix.to_lin hv₁ hv₂) M) (v₁ i) = ∑ (j : m), M j i • v₂ j
@[simp]
theorem
linear_map.to_matrix_id
{R : Type u_1}
[comm_ring R]
{n : Type u_4}
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
[add_comm_group M₁]
[module R M₁]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁) :
⇑(linear_map.to_matrix hv₁ hv₁) linear_map.id = 1
@[simp]
theorem
matrix.to_lin_one
{R : Type u_1}
[comm_ring R]
{n : Type u_4}
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
[add_comm_group M₁]
[module R M₁]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁) :
⇑(matrix.to_lin hv₁ hv₁) 1 = linear_map.id
theorem
linear_map.to_matrix_range
{R : Type u_1}
[comm_ring R]
{m : Type u_3}
{n : Type u_4}
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
[decidable_eq M₁]
[decidable_eq M₂]
(f : M₁ →ₗ[R] M₂)
(k : m)
(i : n) :
⇑(linear_map.to_matrix _ _) f ⟨v₂ k, _⟩ ⟨v₁ i, _⟩ = ⇑(linear_map.to_matrix hv₁ hv₂) f k i
theorem
linear_map.to_matrix_comp
{R : Type u_1}
[comm_ring R]
{l : Type u_2}
{m : Type u_3}
{n : Type u_4}
[fintype l]
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
{M₃ : Type u_7}
[add_comm_group M₃]
[module R M₃]
{v₃ : l → M₃}
(hv₃ : is_basis R v₃)
[decidable_eq m]
(f : M₂ →ₗ[R] M₃)
(g : M₁ →ₗ[R] M₂) :
⇑(linear_map.to_matrix hv₁ hv₃) (f.comp g) = ⇑(linear_map.to_matrix hv₂ hv₃) f ⬝ ⇑(linear_map.to_matrix hv₁ hv₂) g
theorem
linear_map.to_matrix_mul
{R : Type u_1}
[comm_ring R]
{n : Type u_4}
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
[add_comm_group M₁]
[module R M₁]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
(f g : M₁ →ₗ[R] M₁) :
⇑(linear_map.to_matrix hv₁ hv₁) (f * g) = ⇑(linear_map.to_matrix hv₁ hv₁) f ⬝ ⇑(linear_map.to_matrix hv₁ hv₁) g
theorem
matrix.to_lin_mul
{R : Type u_1}
[comm_ring R]
{l : Type u_2}
{m : Type u_3}
{n : Type u_4}
[fintype l]
[fintype m]
[fintype n]
[decidable_eq n]
{M₁ : Type u_5}
{M₂ : Type u_6}
[add_comm_group M₁]
[add_comm_group M₂]
[module R M₁]
[module R M₂]
{v₁ : n → M₁}
(hv₁ : is_basis R v₁)
{v₂ : m → M₂}
(hv₂ : is_basis R v₂)
{M₃ : Type u_7}
[add_comm_group M₃]
[module R M₃]
{v₃ : l → M₃}
(hv₃ : is_basis R v₃)
[decidable_eq m]
(A : matrix l m R)
(B : matrix m n R) :
⇑(matrix.to_lin hv₁ hv₃) (A ⬝ B) = (⇑(matrix.to_lin hv₂ hv₃) A).comp (⇑(matrix.to_lin hv₁ hv₂) B)
def
is_basis.to_matrix
{ι : Type u_1}
{ι' : Type u_2}
[fintype ι]
[fintype ι']
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{e : ι → M}
(he : is_basis R e)
(v : ι' → M) :
matrix ι ι' R
From a basis e : ι → M and a family of vectors v : ι' → M, make the matrix whose columns
are the vectors v i written in the basis e.
theorem
is_basis.to_matrix_eq_to_matrix_constr
{ι : Type u_1}
[fintype ι]
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{e : ι → M}
(he : is_basis R e)
[decidable_eq ι]
(v : ι → M) :
he.to_matrix v = ⇑(linear_map.to_matrix he he) (he.constr v)
@[simp]
theorem
is_basis.to_matrix_self
{ι : Type u_1}
[fintype ι]
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{e : ι → M}
(he : is_basis R e)
[decidable_eq ι] :
theorem
is_basis.to_matrix_update
{ι : Type u_1}
{ι' : Type u_2}
[fintype ι]
[fintype ι']
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{e : ι → M}
(he : is_basis R e)
(v : ι' → M)
(j : ι')
[decidable_eq ι']
(x : M) :
he.to_matrix (function.update v j x) = (he.to_matrix v).update_column j ⇑(⇑(he.repr) x)
@[simp]
theorem
is_basis.to_lin_to_matrix
{ι : Type u_1}
{ι' : Type u_2}
[fintype ι]
[fintype ι']
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{e : ι → M}
(he : is_basis R e)
(v : ι' → M)
[decidable_eq ι']
(hv : is_basis R v) :
⇑(matrix.to_lin hv he) (he.to_matrix v) = linear_map.id
def
is_basis.to_matrix_equiv
{ι : Type u_1}
[fintype ι]
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{e : ι → M}
(he : is_basis R e) :
From a basis e : ι → M, build a linear equivalence between families of vectors v : ι → M,
and matrices, making the matrix whose columns are the vectors v i written in the basis e.
@[simp]
theorem
is_basis_to_matrix_mul_linear_map_to_matrix
{ι : Type u_1}
{ι' : Type u_2}
[fintype ι]
[fintype ι']
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{N : Type u_5}
[add_comm_group N]
[module R N]
{b' : ι' → M}
{c : ι → N}
{c' : ι' → N}
(hb' : is_basis R b')
(hc : is_basis R c)
(hc' : is_basis R c')
(f : M →ₗ[R] N)
[decidable_eq ι'] :
hc.to_matrix c' ⬝ ⇑(linear_map.to_matrix hb' hc') f = ⇑(linear_map.to_matrix hb' hc) f
@[simp]
theorem
linear_map_to_matrix_mul_is_basis_to_matrix
{ι : Type u_1}
{ι' : Type u_2}
[fintype ι]
[fintype ι']
{R : Type u_3}
{M : Type u_4}
[comm_ring R]
[add_comm_group M]
[module R M]
{N : Type u_5}
[add_comm_group N]
[module R N]
{b : ι → M}
{b' : ι' → M}
{c' : ι' → N}
(hb : is_basis R b)
(hb' : is_basis R b')
(hc' : is_basis R c')
(f : M →ₗ[R] N)
[decidable_eq ι]
[decidable_eq ι'] :
⇑(linear_map.to_matrix hb' hc') f ⬝ hb'.to_matrix b = ⇑(linear_map.to_matrix hb hc') f
theorem
linear_equiv.is_unit_det
{R : Type}
[comm_ring R]
{M : Type u_1}
[add_comm_group M]
[module R M]
{M' : Type u_2}
[add_comm_group M']
[module R M']
{ι : Type u_3}
[decidable_eq ι]
[fintype ι]
{v : ι → M}
{v' : ι → M'}
(f : M ≃ₗ[R] M')
(hv : is_basis R v)
(hv' : is_basis R v') :
is_unit (⇑(linear_map.to_matrix hv hv') ↑f).det
def
linear_equiv.of_is_unit_det
{R : Type}
[comm_ring R]
{M : Type u_1}
[add_comm_group M]
[module R M]
{M' : Type u_2}
[add_comm_group M']
[module R M']
{ι : Type u_3}
[decidable_eq ι]
[fintype ι]
{v : ι → M}
{v' : ι → M'}
{f : M →ₗ[R] M'}
{hv : is_basis R v}
{hv' : is_basis R v'}
(h : is_unit (⇑(linear_map.to_matrix hv hv') f).det) :
Builds a linear equivalence from a linear map whose determinant in some bases is a unit.
def
is_basis.det
{R : Type}
[comm_ring R]
{M : Type u_1}
[add_comm_group M]
[module R M]
{ι : Type u_3}
[decidable_eq ι]
[fintype ι]
{e : ι → M}
(he : is_basis R e) :
alternating_map R M R ι
The determinant of a family of vectors with respect to some basis, as an alternating multilinear map.
theorem
is_basis.det_apply
{R : Type}
[comm_ring R]
{M : Type u_1}
[add_comm_group M]
[module R M]
{ι : Type u_3}
[decidable_eq ι]
[fintype ι]
{e : ι → M}
(he : is_basis R e)
(v : ι → M) :
theorem
is_basis.det_self
{R : Type}
[comm_ring R]
{M : Type u_1}
[add_comm_group M]
[module R M]
{ι : Type u_3}
[decidable_eq ι]
[fintype ι]
{e : ι → M}
(he : is_basis R e) :
theorem
is_basis.iff_det
{R : Type}
[comm_ring R]
{M : Type u_1}
[add_comm_group M]
[module R M]
{ι : Type u_3}
[decidable_eq ι]
[fintype ι]
{e : ι → M}
(he : is_basis R e)
{v : ι → M} :
@[simp]
theorem
linear_map.to_matrix_transpose
{K : Type u_1}
{V₁ : Type u_2}
{V₂ : Type u_3}
{ι₁ : Type u_4}
{ι₂ : Type u_5}
[field K]
[add_comm_group V₁]
[vector_space K V₁]
[add_comm_group V₂]
[vector_space K V₂]
[fintype ι₁]
[fintype ι₂]
[decidable_eq ι₁]
[decidable_eq ι₂]
{B₁ : ι₁ → V₁}
(h₁ : is_basis K B₁)
{B₂ : ι₂ → V₂}
(h₂ : is_basis K B₂)
(u : V₁ →ₗ[K] V₂) :
⇑(linear_map.to_matrix _ _) (⇑module.dual.transpose u) = (⇑(linear_map.to_matrix h₁ h₂) u)ᵀ
theorem
linear_map.to_matrix_symm_transpose
{K : Type u_1}
{V₁ : Type u_2}
{V₂ : Type u_3}
{ι₁ : Type u_4}
{ι₂ : Type u_5}
[field K]
[add_comm_group V₁]
[vector_space K V₁]
[add_comm_group V₂]
[vector_space K V₂]
[fintype ι₁]
[fintype ι₂]
[decidable_eq ι₁]
[decidable_eq ι₂]
{B₁ : ι₁ → V₁}
(h₁ : is_basis K B₁)
{B₂ : ι₂ → V₂}
(h₂ : is_basis K B₂)
(M : matrix ι₁ ι₂ K) :
⇑((linear_map.to_matrix _ _).symm) Mᵀ = ⇑module.dual.transpose (⇑(matrix.to_lin h₂ h₁) M)
def
matrix.diag
(n : Type u_2)
[fintype n]
(R : Type v)
(M : Type w)
[semiring R]
[add_comm_monoid M]
[semimodule R M] :
The diagonal of a square matrix.
@[simp]
theorem
matrix.diag_apply
{n : Type u_2}
[fintype n]
{R : Type v}
{M : Type w}
[semiring R]
[add_comm_monoid M]
[semimodule R M]
(A : matrix n n M)
(i : n) :
⇑(matrix.diag n R M) A i = A i i
@[simp]
theorem
matrix.diag_one
{n : Type u_2}
[fintype n]
{R : Type v}
[semiring R]
[decidable_eq n] :
⇑(matrix.diag n R R) 1 = λ (i : n), 1
@[simp]
theorem
matrix.diag_transpose
{n : Type u_2}
[fintype n]
{R : Type v}
{M : Type w}
[semiring R]
[add_comm_monoid M]
[semimodule R M]
(A : matrix n n M) :
⇑(matrix.diag n R M) Aᵀ = ⇑(matrix.diag n R M) A
def
matrix.trace
(n : Type u_2)
[fintype n]
(R : Type v)
(M : Type w)
[semiring R]
[add_comm_monoid M]
[semimodule R M] :
The trace of a square matrix.
Equations
- matrix.trace n R M = {to_fun := λ (A : matrix n n M), ∑ (i : n), ⇑(matrix.diag n R M) A i, map_add' := _, map_smul' := _}
@[simp]
theorem
matrix.trace_diag
{n : Type u_2}
[fintype n]
{R : Type v}
{M : Type w}
[semiring R]
[add_comm_monoid M]
[semimodule R M]
(A : matrix n n M) :
⇑(matrix.trace n R M) A = ∑ (i : n), ⇑(matrix.diag n R M) A i
@[simp]
theorem
matrix.trace_one
{n : Type u_2}
[fintype n]
{R : Type v}
[semiring R]
[decidable_eq n] :
⇑(matrix.trace n R R) 1 = ↑(fintype.card n)
@[simp]
theorem
matrix.trace_transpose
{n : Type u_2}
[fintype n]
{R : Type v}
{M : Type w}
[semiring R]
[add_comm_monoid M]
[semimodule R M]
(A : matrix n n M) :
⇑(matrix.trace n R M) Aᵀ = ⇑(matrix.trace n R M) A
theorem
matrix.trace_mul_comm
{m : Type u_1}
[fintype m]
{n : Type u_2}
[fintype n]
{S : Type v}
[comm_ring S]
(A : matrix m n S)
(B : matrix n m S) :
⇑(matrix.trace n S S) (B ⬝ A) = ⇑(matrix.trace m S S) (A ⬝ B)
theorem
matrix.proj_diagonal
{n : Type u_1}
[fintype n]
[decidable_eq n]
{R : Type v}
[comm_ring R]
(i : n)
(w : n → R) :
(linear_map.proj i).comp (⇑matrix.to_lin' (matrix.diagonal w)) = w i • linear_map.proj i
theorem
matrix.diagonal_comp_std_basis
{n : Type u_1}
[fintype n]
[decidable_eq n]
{R : Type v}
[comm_ring R]
(w : n → R)
(i : n) :
(⇑matrix.to_lin' (matrix.diagonal w)).comp (linear_map.std_basis R (λ (_x : n), R) i) = w i • linear_map.std_basis R (λ (_x : n), R) i
theorem
matrix.diagonal_to_lin'
{n : Type u_1}
[fintype n]
[decidable_eq n]
{R : Type v}
[comm_ring R]
(w : n → R) :
⇑matrix.to_lin' (matrix.diagonal w) = linear_map.pi (λ (i : n), w i • linear_map.proj i)
def
matrix.to_linear_equiv
{n : Type u_1}
[fintype n]
[decidable_eq n]
{R : Type v}
[comm_ring R]
(P : matrix n n R)
(h : is_unit P) :
An invertible matrix yields a linear equivalence from the free module to itself.
@[simp]
theorem
matrix.to_linear_equiv_apply
{n : Type u_1}
[fintype n]
[decidable_eq n]
{R : Type v}
[comm_ring R]
(P : matrix n n R)
(h : is_unit P) :
↑(P.to_linear_equiv h) = ⇑matrix.to_lin' P
@[simp]
theorem
matrix.to_linear_equiv_symm_apply
{n : Type u_1}
[fintype n]
[decidable_eq n]
{R : Type v}
[comm_ring R]
(P : matrix n n R)
(h : is_unit P) :
↑((P.to_linear_equiv h).symm) = ⇑matrix.to_lin' P⁻¹
theorem
matrix.rank_vec_mul_vec
{K : Type u}
[field K]
{m n : Type u}
[fintype m]
[fintype n]
[decidable_eq n]
(w : m → K)
(v : n → K) :
rank (⇑matrix.to_lin' (matrix.vec_mul_vec w v)) ≤ 1
theorem
matrix.ker_diagonal_to_lin'
{m : Type u_1}
[fintype m]
{K : Type u}
[field K]
[decidable_eq m]
(w : m → K) :
(⇑matrix.to_lin' (matrix.diagonal w)).ker = ⨆ (i : m) (H : i ∈ {i : m | w i = 0}), (linear_map.std_basis K (λ (i : m), K) i).range
theorem
matrix.range_diagonal
{m : Type u_1}
[fintype m]
{K : Type u}
[field K]
[decidable_eq m]
(w : m → K) :
(⇑matrix.to_lin' (matrix.diagonal w)).range = ⨆ (i : m) (H : i ∈ {i : m | w i ≠ 0}), (linear_map.std_basis K (λ (i : m), K) i).range
theorem
matrix.rank_diagonal
{m : Type u_1}
[fintype m]
{K : Type u}
[field K]
[decidable_eq m]
[decidable_eq K]
(w : m → K) :
rank (⇑matrix.to_lin' (matrix.diagonal w)) = ↑(fintype.card {i // w i ≠ 0})
@[instance]
def
matrix.finite_dimensional
{m : Type u_1}
{n : Type u_2}
[fintype m]
[fintype n]
{R : Type v}
[field R] :
finite_dimensional R (matrix m n R)
@[simp]
theorem
matrix.findim_matrix
{m : Type u_1}
{n : Type u_2}
[fintype m]
[fintype n]
{R : Type v}
[field R] :
finite_dimensional.findim R (matrix m n R) = (fintype.card m) * fintype.card n
The dimension of the space of finite dimensional matrices is the product of the number of rows and columns.
def
matrix.reindex_linear_equiv
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{m' : Type u_5}
{n' : Type u_6}
[fintype m']
[fintype n']
{R : Type v}
[semiring R]
(eₘ : m ≃ m')
(eₙ : n ≃ n') :
The natural map that reindexes a matrix's rows and columns with equivalent types is a linear equivalence.
Equations
- matrix.reindex_linear_equiv eₘ eₙ = {to_fun := (matrix.reindex eₘ eₙ).to_fun, map_add' := _, map_smul' := _, inv_fun := (matrix.reindex eₘ eₙ).inv_fun, left_inv := _, right_inv := _}
theorem
matrix.reindex_mul
{l : Type u_1}
{m : Type u_2}
{n : Type u_3}
[fintype l]
[fintype m]
[fintype n]
{l' : Type u_4}
{m' : Type u_5}
{n' : Type u_6}
[fintype l']
[fintype m']
[fintype n']
{R : Type v}
[semiring R]
(eₘ : m ≃ m')
(eₙ : n ≃ n')
(eₗ : l ≃ l')
(M : matrix m n R)
(N : matrix n l R) :
⇑(matrix.reindex_linear_equiv eₘ eₙ) M ⬝ ⇑(matrix.reindex_linear_equiv eₙ eₗ) N = ⇑(matrix.reindex_linear_equiv eₘ eₗ) (M ⬝ N)
def
matrix.reindex_alg_equiv
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{R : Type v}
[comm_semiring R]
[decidable_eq m]
[decidable_eq n]
(e : m ≃ n) :
For square matrices, the natural map that reindexes a matrix's rows and columns with equivalent types is an equivalence of algebras.
Equations
- matrix.reindex_alg_equiv e = {to_fun := (matrix.reindex_linear_equiv e e).to_fun, inv_fun := (matrix.reindex_linear_equiv e e).inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _, commutes' := _}
@[simp]
theorem
matrix.reindex_alg_equiv_apply
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{R : Type v}
[comm_semiring R]
[decidable_eq m]
[decidable_eq n]
(e : m ≃ n)
(M : matrix m m R) :
@[simp]
theorem
matrix.reindex_alg_equiv_symm_apply
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{R : Type v}
[comm_semiring R]
[decidable_eq m]
[decidable_eq n]
(e : m ≃ n)
(M : matrix n n R) :
@[simp]
theorem
matrix.det_reindex_self'
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{R : Type v}
[decidable_eq m]
[decidable_eq n]
[comm_ring R]
(e : m ≃ n)
(A : matrix m m R) :
simp version of det_reindex_self
det_reindex_self is not a good simp lemma because reindex_apply fires before.
So we have this lemma to continue from there.
theorem
matrix.det_reindex_self
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{R : Type v}
[decidable_eq m]
[decidable_eq n]
[comm_ring R]
(e : m ≃ n)
(A : matrix m m R) :
(⇑(matrix.reindex e e) A).det = A.det
Reindexing both indices along the same equivalence preserves the determinant.
For the simp version of this lemma, see det_reindex_self'.
theorem
matrix.det_reindex_linear_equiv_self
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{R : Type v}
[decidable_eq m]
[decidable_eq n]
[comm_ring R]
(e : m ≃ n)
(A : matrix m m R) :
(⇑(matrix.reindex_linear_equiv e e) A).det = A.det
Reindexing both indices along the same equivalence preserves the determinant.
For the simp version of this lemma, see det_reindex_self'.
theorem
matrix.det_reindex_alg_equiv
{m : Type u_2}
{n : Type u_3}
[fintype m]
[fintype n]
{R : Type v}
[decidable_eq m]
[decidable_eq n]
[comm_ring R]
(e : m ≃ n)
(A : matrix m m R) :
(⇑(matrix.reindex_alg_equiv e) A).det = A.det
Reindexing both indices along the same equivalence preserves the determinant.
For the simp version of this lemma, see det_reindex_self'.
def
linear_map.trace_aux
(R : Type u)
[comm_ring R]
{M : Type v}
[add_comm_group M]
[module R M]
{ι : Type w}
[decidable_eq ι]
[fintype ι]
{b : ι → M}
(hb : is_basis R b) :
The trace of an endomorphism given a basis.
Equations
- linear_map.trace_aux R hb = (matrix.trace ι R R).comp ↑(linear_map.to_matrix hb hb)
@[simp]
theorem
linear_map.trace_aux_def
(R : Type u)
[comm_ring R]
{M : Type v}
[add_comm_group M]
[module R M]
{ι : Type w}
[decidable_eq ι]
[fintype ι]
{b : ι → M}
(hb : is_basis R b)
(f : M →ₗ[R] M) :
⇑(linear_map.trace_aux R hb) f = ⇑(matrix.trace ι R R) (⇑(linear_map.to_matrix hb hb) f)
theorem
linear_map.trace_aux_eq'
(R : Type u)
[comm_ring R]
{M : Type v}
[add_comm_group M]
[module R M]
{ι : Type w}
[decidable_eq ι]
[fintype ι]
{b : ι → M}
(hb : is_basis R b)
{κ : Type w}
[decidable_eq κ]
[fintype κ]
{c : κ → M}
(hc : is_basis R c) :
linear_map.trace_aux R hb = linear_map.trace_aux R hc
theorem
linear_map.trace_aux_range
(R : Type u)
[comm_ring R]
{M : Type v}
[add_comm_group M]
[module R M]
{ι : Type w}
[decidable_eq ι]
[fintype ι]
{b : ι → M}
(hb : is_basis R b) :
linear_map.trace_aux R _ = linear_map.trace_aux R hb
theorem
linear_map.trace_aux_eq
(R : Type u)
[comm_ring R]
{M : Type v}
[add_comm_group M]
[module R M]
{ι : Type u_1}
[decidable_eq ι]
[fintype ι]
{b : ι → M}
(hb : is_basis R b)
{κ : Type u_2}
[decidable_eq κ]
[fintype κ]
{c : κ → M}
(hc : is_basis R c) :
linear_map.trace_aux R hb = linear_map.trace_aux R hc
where ι and κ can reside in different universes
Trace of an endomorphism independent of basis.
theorem
linear_map.trace_eq_matrix_trace
(R : Type u)
[comm_ring R]
{M : Type v}
[add_comm_group M]
[module R M]
{ι : Type w}
[fintype ι]
[decidable_eq ι]
{b : ι → M}
(hb : is_basis R b)
(f : M →ₗ[R] M) :
⇑(linear_map.trace R M) f = ⇑(matrix.trace ι R R) (⇑(linear_map.to_matrix hb hb) f)
theorem
linear_map.trace_mul_comm
(R : Type u)
[comm_ring R]
{M : Type v}
[add_comm_group M]
[module R M]
(f g : M →ₗ[R] M) :
⇑(linear_map.trace R M) (f * g) = ⇑(linear_map.trace R M) (g * f)
@[instance]
def
linear_map.finite_dimensional
{K : Type u_1}
[field K]
{V : Type u_2}
[add_comm_group V]
[vector_space K V]
[finite_dimensional K V]
{W : Type u_3}
[add_comm_group W]
[vector_space K W]
[finite_dimensional K W] :
finite_dimensional K (V →ₗ[K] W)
@[simp]
theorem
linear_map.findim_linear_map
{K : Type u_1}
[field K]
{V : Type u_2}
[add_comm_group V]
[vector_space K V]
[finite_dimensional K V]
{W : Type u_3}
[add_comm_group W]
[vector_space K W]
[finite_dimensional K W] :
finite_dimensional.findim K (V →ₗ[K] W) = (finite_dimensional.findim K V) * finite_dimensional.findim K W
The dimension of the space of linear transformations is the product of the dimensions of the domain and codomain.
def
alg_equiv_matrix'
{R : Type v}
[comm_ring R]
{n : Type u_1}
[fintype n]
[decidable_eq n] :
module.End R (n → R) ≃ₐ[R] matrix n n R
The natural equivalence between linear endomorphisms of finite free modules and square matrices is compatible with the algebra structures.
Equations
- alg_equiv_matrix' = {to_fun := linear_map.to_matrix'.to_fun, inv_fun := linear_map.to_matrix'.inv_fun, left_inv := _, right_inv := _, map_mul' := _, map_add' := _, commutes' := _}
def
linear_equiv.alg_conj
{R : Type v}
[comm_ring R]
{M₁ : Type u_1}
{M₂ : Type (max u_2 u_3)}
[add_comm_group M₁]
[module R M₁]
[add_comm_group M₂]
[module R M₂]
(e : M₁ ≃ₗ[R] M₂) :
module.End R M₁ ≃ₐ[R] module.End R M₂
A linear equivalence of two modules induces an equivalence of algebras of their endomorphisms.
def
alg_equiv_matrix
{R : Type v}
{M : Type w}
{n : Type u_1}
[fintype n]
[comm_ring R]
[add_comm_group M]
[module R M]
[decidable_eq n]
{b : n → M}
(h : is_basis R b) :
module.End R M ≃ₐ[R] matrix n n R
A basis of a module induces an equivalence of algebras from the endomorphisms of the module to square matrices.
Equations
@[simp]
theorem
matrix.dot_product_std_basis_eq_mul
{R : Type v}
[semiring R]
{n : Type w}
[fintype n]
[decidable_eq n]
(v : n → R)
(c : R)
(i : n) :
matrix.dot_product v (⇑(linear_map.std_basis R (λ (_x : n), R) i) c) = (v i) * c
@[simp]
theorem
matrix.dot_product_std_basis_one
{R : Type v}
[semiring R]
{n : Type w}
[fintype n]
[decidable_eq n]
(v : n → R)
(i : n) :
matrix.dot_product v (⇑(linear_map.std_basis R (λ (_x : n), R) i) 1) = v i
theorem
matrix.dot_product_eq
{R : Type v}
[semiring R]
{n : Type w}
[fintype n]
(v w : n → R)
(h : ∀ (u : n → R), matrix.dot_product v u = matrix.dot_product w u) :
v = w
theorem
matrix.dot_product_eq_iff
{R : Type v}
[semiring R]
{n : Type w}
[fintype n]
{v w : n → R} :
(∀ (u : n → R), matrix.dot_product v u = matrix.dot_product w u) ↔ v = w
theorem
matrix.dot_product_eq_zero
{R : Type v}
[semiring R]
{n : Type w}
[fintype n]
(v : n → R)
(h : ∀ (w : n → R), matrix.dot_product v w = 0) :
v = 0
theorem
matrix.dot_product_eq_zero_iff
{R : Type v}
[semiring R]
{n : Type w}
[fintype n]
{v : n → R} :
(∀ (w : n → R), matrix.dot_product v w = 0) ↔ v = 0