mathlib documentation

analysis.inner_product_space.pi_L2

inner product space structure on finite products of inner product spaces #

The norm on a finite product of inner product spaces is compatible with an inner product $$ \langle x, y\rangle = \sum \langle x_i, y_i \rangle. $$ This is recorded in this file as an inner product space instance on pi_Lp 2.

This file develops the notion of a finite dimensional Hilbert space over 𝕜 = ℂ, ℝ, referred to as E. We define an orthonormal_basis 𝕜 ι E as a linear isometric equivalence between E and euclidean_space 𝕜 ι. Then std_orthonormal_basis shows that such an equivalence always exists if E is finite dimensional. We provide language for converting between a basis that is orthonormal and an orthonormal basis (e.g. basis.to_orthonormal_basis). We show that orthonormal bases for each summand in a direct sum of spaces can be combined into an orthonormal basis for the the whole sum in direct_sum.submodule_is_internal.subordinate_orthonormal_basis. In the last section, various properties of matrices are explored.

Main definitions #

For consequences in infinite dimension (Hilbert bases, etc.), see the file analysis.inner_product_space.l2_space.

@[protected, instance]
noncomputable def pi_Lp.inner_product_space {𝕜 : Type u_3} [is_R_or_C 𝕜] {ι : Type u_1} [fintype ι] (f : ι → Type u_2) [Π (i : ι), inner_product_space 𝕜 (f i)] :
Equations
@[simp]
theorem pi_Lp.inner_apply {𝕜 : Type u_3} [is_R_or_C 𝕜] {ι : Type u_1} [fintype ι] {f : ι → Type u_2} [Π (i : ι), inner_product_space 𝕜 (f i)] (x y : pi_Lp 2 f) :
has_inner.inner x y = finset.univ.sum (λ (i : ι), has_inner.inner (x i) (y i))
@[nolint, reducible]
def euclidean_space (𝕜 : Type u_1) [is_R_or_C 𝕜] (n : Type u_2) [fintype n] :
Type (max u_2 u_1)

The standard real/complex Euclidean space, functions on a finite type. For an n-dimensional space use euclidean_space 𝕜 (fin n).

Equations
theorem euclidean_space.nnnorm_eq {𝕜 : Type u_1} [is_R_or_C 𝕜] {n : Type u_2} [fintype n] (x : euclidean_space 𝕜 n) :
theorem euclidean_space.norm_eq {𝕜 : Type u_1} [is_R_or_C 𝕜] {n : Type u_2} [fintype n] (x : euclidean_space 𝕜 n) :
x = real.sqrt (finset.univ.sum (λ (i : n), x i ^ 2))
theorem euclidean_space.dist_eq {𝕜 : Type u_1} [is_R_or_C 𝕜] {n : Type u_2} [fintype n] (x y : euclidean_space 𝕜 n) :
has_dist.dist x y = real.sqrt (finset.univ.sum (λ (i : n), has_dist.dist (x i) (y i) ^ 2))
theorem euclidean_space.nndist_eq {𝕜 : Type u_1} [is_R_or_C 𝕜] {n : Type u_2} [fintype n] (x y : euclidean_space 𝕜 n) :
theorem euclidean_space.edist_eq {𝕜 : Type u_1} [is_R_or_C 𝕜] {n : Type u_2} [fintype n] (x y : euclidean_space 𝕜 n) :
has_edist.edist x y = finset.univ.sum (λ (i : n), has_edist.edist (x i) (y i) ^ 2) ^ (1 / 2)
@[protected, instance]
def euclidean_space.finite_dimensional {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] :
@[protected, instance]
noncomputable def euclidean_space.inner_product_space {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] :
Equations
@[simp]
theorem finrank_euclidean_space {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] :
theorem finrank_euclidean_space_fin {𝕜 : Type u_3} [is_R_or_C 𝕜] {n : } :
theorem euclidean_space.inner_eq_star_dot_product {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] (x y : euclidean_space 𝕜 ι) :
has_inner.inner x y = matrix.dot_product (has_star.star ((pi_Lp.equiv 2 (λ (i : ι), 𝕜)) x)) ((pi_Lp.equiv 2 (λ (i : ι), 𝕜)) y)
noncomputable def direct_sum.is_internal.isometry_L2_of_orthogonal_family {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V) (hV' : orthogonal_family 𝕜 (λ (i : ι), (V i).subtypeₗᵢ)) :
E ≃ₗᵢ[𝕜] pi_Lp 2 (λ (i : ι), (V i))

A finite, mutually orthogonal family of subspaces of E, which span E, induce an isometry from E to pi_Lp 2 of the subspaces equipped with the L2 inner product.

Equations
@[simp]
theorem direct_sum.is_internal.isometry_L2_of_orthogonal_family_symm_apply {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V) (hV' : orthogonal_family 𝕜 (λ (i : ι), (V i).subtypeₗᵢ)) (w : pi_Lp 2 (λ (i : ι), (V i))) :
((hV.isometry_L2_of_orthogonal_family hV').symm) w = finset.univ.sum (λ (i : ι), (w i))
noncomputable def euclidean_space.equiv (ι : Type u_1) (𝕜 : Type u_3) [is_R_or_C 𝕜] [fintype ι] :
euclidean_space 𝕜 ι ≃L[𝕜] ι → 𝕜

pi_Lp.linear_equiv upgraded to a continuous linear map between euclidean_space 𝕜 ι and ι → 𝕜.

Equations
@[simp]
theorem euclidean_space.equiv_to_linear_equiv_apply (ι : Type u_1) (𝕜 : Type u_3) [is_R_or_C 𝕜] [fintype ι] (ᾰ : pi_Lp 2 (λ (i : ι), 𝕜)) (i : ι) :
@[simp]
theorem euclidean_space.equiv_apply (ι : Type u_1) (𝕜 : Type u_3) [is_R_or_C 𝕜] [fintype ι] (ᾰ : pi_Lp 2 (λ (i : ι), 𝕜)) (i : ι) :
(euclidean_space.equiv ι 𝕜) i = ᾰ i
@[simp]
theorem euclidean_space.equiv_symm_apply (ι : Type u_1) (𝕜 : Type u_3) [is_R_or_C 𝕜] [fintype ι] (ᾰ : Π (i : ι), (λ (i : ι), 𝕜) i) :
((euclidean_space.equiv ι 𝕜).symm) =
@[simp]
theorem euclidean_space.equiv_to_linear_equiv_symm_apply (ι : Type u_1) (𝕜 : Type u_3) [is_R_or_C 𝕜] [fintype ι] (ᾰ : Π (i : ι), (λ (i : ι), 𝕜) i) :
noncomputable def euclidean_space.projₗ {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] (i : ι) :
euclidean_space 𝕜 ι →ₗ[𝕜] 𝕜

The projection on the i-th coordinate of euclidean_space 𝕜 ι, as a linear map.

Equations
@[simp]
theorem euclidean_space.projₗ_apply {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] (i : ι) (ᾰ : euclidean_space 𝕜 ι) :
@[simp]
theorem euclidean_space.proj_coe {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] (i : ι) :
@[simp]
theorem euclidean_space.proj_apply {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] (i : ι) (ᾰ : euclidean_space 𝕜 ι) :
noncomputable def euclidean_space.proj {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] (i : ι) :
euclidean_space 𝕜 ι →L[𝕜] 𝕜

The projection on the i-th coordinate of euclidean_space 𝕜 ι, as a continuous linear map.

Equations
noncomputable def euclidean_space.single {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] [decidable_eq ι] (i : ι) (a : 𝕜) :

The vector given in euclidean space by being 1 : 𝕜 at coordinate i : ι and 0 : 𝕜 at all other coordinates.

Equations
@[simp]
theorem pi_Lp.equiv_single {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] [decidable_eq ι] (i : ι) (a : 𝕜) :
(pi_Lp.equiv 2 (λ (i : ι), 𝕜)) (euclidean_space.single i a) = pi.single i a
@[simp]
theorem pi_Lp.equiv_symm_single {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] [decidable_eq ι] (i : ι) (a : 𝕜) :
((pi_Lp.equiv 2 (λ (i : ι), 𝕜)).symm) (pi.single i a) = euclidean_space.single i a
@[simp]
theorem euclidean_space.single_apply {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] [decidable_eq ι] (i : ι) (a : 𝕜) (j : ι) :
euclidean_space.single i a j = ite (j = i) a 0
theorem euclidean_space.inner_single_left {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] [decidable_eq ι] (i : ι) (a : 𝕜) (v : euclidean_space 𝕜 ι) :
theorem euclidean_space.inner_single_right {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] [decidable_eq ι] (i : ι) (a : 𝕜) (v : euclidean_space 𝕜 ι) :
has_inner.inner v (euclidean_space.single i a) = a * (star_ring_end ((λ (i : ι), 𝕜) i)) (v i)
theorem euclidean_space.pi_Lp_congr_left_single {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] [decidable_eq ι] {ι' : Type u_2} [fintype ι'] [decidable_eq ι'] (e : ι' ι) (i' : ι') :
structure orthonormal_basis (ι : Type u_1) (𝕜 : Type u_3) [is_R_or_C 𝕜] (E : Type u_4) [inner_product_space 𝕜 E] [fintype ι] :
Type (max u_1 u_3 u_4)

An orthonormal basis on E is an identification of E with its dimensional-matching euclidean_space 𝕜 ι.

Instances for orthonormal_basis
@[protected, instance]
noncomputable def orthonormal_basis.inhabited {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] [fintype ι] :
Equations
@[protected, instance]
noncomputable def orthonormal_basis.has_coe_to_fun {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] :
has_coe_to_fun (orthonormal_basis ι 𝕜 E) (λ (_x : orthonormal_basis ι 𝕜 E), ι → E)

b i is the ith basis vector.

Equations
@[simp]
theorem orthonormal_basis.coe_of_repr {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [decidable_eq ι] (e : E ≃ₗᵢ[𝕜] euclidean_space 𝕜 ι) :
{repr := e} = λ (i : ι), (e.symm) (euclidean_space.single i 1)
@[protected, simp]
theorem orthonormal_basis.repr_symm_single {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [decidable_eq ι] (b : orthonormal_basis ι 𝕜 E) (i : ι) :
@[protected, simp]
theorem orthonormal_basis.repr_self {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [decidable_eq ι] (b : orthonormal_basis ι 𝕜 E) (i : ι) :
@[protected]
theorem orthonormal_basis.repr_apply_apply {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) (v : E) (i : ι) :
(b.repr) v i = has_inner.inner (b i) v
@[protected, simp]
theorem orthonormal_basis.orthonormal {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) :
@[protected]
noncomputable def orthonormal_basis.to_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) :
basis ι 𝕜 E

The basis ι 𝕜 E underlying the orthonormal_basis -

Equations
@[protected, simp]
theorem orthonormal_basis.coe_to_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) :
@[protected, simp]
theorem orthonormal_basis.coe_to_basis_repr {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) :
@[protected, simp]
theorem orthonormal_basis.coe_to_basis_repr_apply {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) (x : E) (i : ι) :
((b.to_basis.repr) x) i = (b.repr) x i
@[protected]
theorem orthonormal_basis.sum_repr {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) (x : E) :
finset.univ.sum (λ (i : ι), (b.repr) x i b i) = x
@[protected]
theorem orthonormal_basis.sum_repr_symm {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) (v : euclidean_space 𝕜 ι) :
finset.univ.sum (λ (i : ι), v i b i) = (b.repr.symm) v
@[protected]
theorem orthonormal_basis.sum_inner_mul_inner {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (b : orthonormal_basis ι 𝕜 E) (x y : E) :
@[protected]
theorem orthonormal_basis.orthogonal_projection_eq_sum {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {U : submodule 𝕜 E} [complete_space U] (b : orthonormal_basis ι 𝕜 U) (x : E) :
@[protected]
noncomputable def orthonormal_basis.map {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {G : Type u_2} [inner_product_space 𝕜 G] (b : orthonormal_basis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) :

Mapping an orthonormal basis along a linear_isometry_equiv.

Equations
@[protected, simp]
theorem orthonormal_basis.map_apply {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {G : Type u_2} [inner_product_space 𝕜 G] (b : orthonormal_basis ι 𝕜 E) (L : E ≃ₗᵢ[𝕜] G) (i : ι) :
(b.map L) i = L (b i)
noncomputable def basis.to_orthonormal_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :

A basis that is orthonormal is an orthonormal basis.

Equations
@[simp]
theorem basis.coe_to_orthonormal_basis_repr {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
@[simp]
theorem basis.coe_to_orthonormal_basis_repr_symm {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
@[simp]
theorem basis.to_basis_to_orthonormal_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
@[simp]
theorem basis.coe_to_orthonormal_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) :
@[protected]
noncomputable def orthonormal_basis.mk {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {v : ι → E} (hon : orthonormal 𝕜 v) (hsp : submodule.span 𝕜 (set.range v)) :

A finite orthonormal set that spans is an orthonormal basis

Equations
@[protected, simp]
theorem orthonormal_basis.coe_mk {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {v : ι → E} (hon : orthonormal 𝕜 v) (hsp : submodule.span 𝕜 (set.range v)) :
@[protected]
noncomputable def orthonormal_basis.span {ι' : Type u_2} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] {v' : ι' → E} (h : orthonormal 𝕜 v') (s : finset ι') :

Any finite subset of a orthonormal family is an orthonormal_basis for its span.

Equations
@[protected, simp]
theorem orthonormal_basis.span_apply {ι' : Type u_2} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] {v' : ι' → E} (h : orthonormal 𝕜 v') (s : finset ι') (i : s) :
@[protected]
noncomputable def orthonormal_basis.mk_of_orthogonal_eq_bot {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {v : ι → E} (hon : orthonormal 𝕜 v) (hsp : (submodule.span 𝕜 (set.range v)) = ) :

A finite orthonormal family of vectors whose span has trivial orthogonal complement is an orthonormal basis.

Equations
@[protected, simp]
theorem orthonormal_basis.coe_of_orthogonal_eq_bot_mk {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {v : ι → E} (hon : orthonormal 𝕜 v) (hsp : (submodule.span 𝕜 (set.range v)) = ) :
noncomputable def orthonormal_basis.reindex {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [fintype ι'] (b : orthonormal_basis ι 𝕜 E) (e : ι ι') :

b.reindex (e : ι ≃ ι') is an orthonormal_basis indexed by ι'

Equations
@[protected]
theorem orthonormal_basis.reindex_apply {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [fintype ι'] (b : orthonormal_basis ι 𝕜 E) (e : ι ι') (i' : ι') :
(b.reindex e) i' = b ((e.symm) i')
@[protected, simp]
theorem orthonormal_basis.coe_reindex {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [fintype ι'] (b : orthonormal_basis ι 𝕜 E) (e : ι ι') :
@[protected, simp]
theorem orthonormal_basis.reindex_repr {ι : Type u_1} {ι' : Type u_2} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [fintype ι'] (b : orthonormal_basis ι 𝕜 E) (e : ι ι') (x : E) (i' : ι') :
((b.reindex e).repr) x i' = (b.repr) x ((e.symm) i')
@[simp]
theorem basis.map_isometry_euclidean_of_orthonormal {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] {E' : Type u_5} [inner_product_space 𝕜 E'] [fintype ι] (v : basis ι 𝕜 E) (hv : orthonormal 𝕜 v) (f : E ≃ₗᵢ[𝕜] E') :

If f : E ≃ₗᵢ[𝕜] E' is a linear isometry of inner product spaces then an orthonormal basis v of E determines a linear isometry e : E' ≃ₗᵢ[𝕜] euclidean_space 𝕜 ι. This result states that e may be obtained either by transporting v to E' or by composing with the linear isometry E ≃ₗᵢ[𝕜] euclidean_space 𝕜 ι provided by v.

is isometric to ℝ² with the Euclidean inner product.

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noncomputable def complex.isometry_of_orthonormal {F : Type u_6} [inner_product_space F] {v : basis (fin 2) F} (hv : orthonormal v) :

The isometry between and a two-dimensional real inner product space given by a basis.

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Matrix representation of an orthonormal basis with respect to another #

theorem orthonormal_basis.to_matrix_orthonormal_basis_mem_unitary {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [decidable_eq ι] (a b : orthonormal_basis ι 𝕜 E) :

The change-of-basis matrix between two orthonormal bases a, b is a unitary matrix.

@[simp]
theorem orthonormal_basis.det_to_matrix_orthonormal_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [decidable_eq ι] (a b : orthonormal_basis ι 𝕜 E) :

The determinant of the change-of-basis matrix between two orthonormal bases a, b has unit length.

The change-of-basis matrix between two orthonormal bases a, b is an orthogonal matrix.

The determinant of the change-of-basis matrix between two orthonormal bases a, b is ±1.

Existence of orthonormal basis, etc. #

noncomputable def direct_sum.is_internal.collected_orthonormal_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {A : ι → submodule 𝕜 E} (hV : orthogonal_family 𝕜 (λ (i : ι), (A i).subtypeₗᵢ)) [decidable_eq ι] (hV_sum : direct_sum.is_internal (λ (i : ι), A i)) {α : ι → Type u_2} [Π (i : ι), fintype (α i)] (v_family : Π (i : ι), orthonormal_basis (α i) 𝕜 (A i)) :
orthonormal_basis (Σ (i : ι), α i) 𝕜 E

Given an internal direct sum decomposition of a module M, and an orthonormal basis for each of the components of the direct sum, the disjoint union of these orthonormal bases is an orthonormal basis for M.

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theorem direct_sum.is_internal.collected_orthonormal_basis_mem {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] {A : ι → submodule 𝕜 E} [decidable_eq ι] (h : direct_sum.is_internal A) {α : ι → Type u_2} [Π (i : ι), fintype (α i)] (hV : orthogonal_family 𝕜 (λ (i : ι), (A i).subtypeₗᵢ)) (v : Π (i : ι), orthonormal_basis (α i) 𝕜 (A i)) (a : Σ (i : ι), α i) :
theorem orthonormal.exists_orthonormal_basis_extension {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] {v : set E} [finite_dimensional 𝕜 E] (hv : orthonormal 𝕜 coe) :
∃ {u : finset E} (b : orthonormal_basis u 𝕜 E), v u b = coe

In a finite-dimensional inner_product_space, any orthonormal subset can be extended to an orthonormal basis.

theorem exists_orthonormal_basis (𝕜 : Type u_3) [is_R_or_C 𝕜] (E : Type u_4) [inner_product_space 𝕜 E] [finite_dimensional 𝕜 E] :
∃ (w : finset E) (b : orthonormal_basis w 𝕜 E), b = coe

A finite-dimensional inner product space admits an orthonormal basis.

noncomputable def orthonormal_basis_index (𝕜 : Type u_3) [is_R_or_C 𝕜] (E : Type u_4) [inner_product_space 𝕜 E] [finite_dimensional 𝕜 E] :

Index for an arbitrary orthonormal basis on a finite-dimensional inner_product_space.

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noncomputable def std_orthonormal_basis (𝕜 : Type u_3) [is_R_or_C 𝕜] (E : Type u_4) [inner_product_space 𝕜 E] [finite_dimensional 𝕜 E] :

A finite-dimensional inner_product_space has an orthonormal basis.

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@[simp]
theorem coe_std_orthonormal_basis (𝕜 : Type u_3) [is_R_or_C 𝕜] (E : Type u_4) [inner_product_space 𝕜 E] [finite_dimensional 𝕜 E] :
noncomputable def fin_std_orthonormal_basis {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [finite_dimensional 𝕜 E] {n : } (hn : finite_dimensional.finrank 𝕜 E = n) :

An n-dimensional inner_product_space has an orthonormal basis indexed by fin n.

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@[irreducible]
noncomputable def direct_sum.is_internal.sigma_orthonormal_basis_index_equiv {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [finite_dimensional 𝕜 E] {n : } (hn : finite_dimensional.finrank 𝕜 E = n) [decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V) (hV' : orthogonal_family 𝕜 (λ (i : ι), (V i).subtypeₗᵢ)) :
(Σ (i : ι), (orthonormal_basis_index 𝕜 (V i))) fin n

Exhibit a bijection between fin n and the index set of a certain basis of an n-dimensional inner product space E. This should not be accessed directly, but only via the subsequent API.

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@[irreducible]
noncomputable def direct_sum.is_internal.subordinate_orthonormal_basis {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [finite_dimensional 𝕜 E] {n : } (hn : finite_dimensional.finrank 𝕜 E = n) [decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V) (hV' : orthogonal_family 𝕜 (λ (i : ι), (V i).subtypeₗᵢ)) :

An n-dimensional inner_product_space equipped with a decomposition as an internal direct sum has an orthonormal basis indexed by fin n and subordinate to that direct sum.

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@[irreducible]
noncomputable def direct_sum.is_internal.subordinate_orthonormal_basis_index {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [finite_dimensional 𝕜 E] {n : } (hn : finite_dimensional.finrank 𝕜 E = n) [decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V) (a : fin n) (hV' : orthogonal_family 𝕜 (λ (i : ι), (V i).subtypeₗᵢ)) :
ι

An n-dimensional inner_product_space equipped with a decomposition as an internal direct sum has an orthonormal basis indexed by fin n and subordinate to that direct sum. This function provides the mapping by which it is subordinate.

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theorem direct_sum.is_internal.subordinate_orthonormal_basis_subordinate {ι : Type u_1} {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] [fintype ι] [finite_dimensional 𝕜 E] {n : } (hn : finite_dimensional.finrank 𝕜 E = n) [decidable_eq ι] {V : ι → submodule 𝕜 E} (hV : direct_sum.is_internal V) (a : fin n) (hV' : orthogonal_family 𝕜 (λ (i : ι), (V i).subtypeₗᵢ)) :

The basis constructed in orthogonal_family.subordinate_orthonormal_basis is subordinate to the orthogonal_family in question.

noncomputable def orthonormal_basis.from_orthogonal_span_singleton {𝕜 : Type u_3} [is_R_or_C 𝕜] {E : Type u_4} [inner_product_space 𝕜 E] (n : ) [fact (finite_dimensional.finrank 𝕜 E = n + 1)] {v : E} (hv : v 0) :

Given a natural number n one less than the finrank of a finite-dimensional inner product space, there exists an isometry from the orthogonal complement of a nonzero singleton to euclidean_space 𝕜 (fin n).

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noncomputable def linear_isometry.extend {𝕜 : Type u_3} [is_R_or_C 𝕜] {V : Type u_8} [inner_product_space 𝕜 V] [finite_dimensional 𝕜 V] {S : submodule 𝕜 V} (L : S →ₗᵢ[𝕜] V) :
V →ₗᵢ[𝕜] V

Let S be a subspace of a finite-dimensional complex inner product space V. A linear isometry mapping S into V can be extended to a full isometry of V.

TODO: The case when S is a finite-dimensional subspace of an infinite-dimensional V.

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theorem linear_isometry.extend_apply {𝕜 : Type u_3} [is_R_or_C 𝕜] {V : Type u_8} [inner_product_space 𝕜 V] [finite_dimensional 𝕜 V] {S : submodule 𝕜 V} (L : S →ₗᵢ[𝕜] V) (s : S) :
(L.extend) s = L s
theorem inner_matrix_row_row {𝕜 : Type u_3} [is_R_or_C 𝕜] {n m : } (A B : matrix (fin n) (fin m) 𝕜) (i j : fin n) :
has_inner.inner (A i) (B j) = B.mul A.conj_transpose j i

The inner product of a row of A and a row of B is an entry of B ⬝ Aᴴ.

theorem inner_matrix_col_col {𝕜 : Type u_3} [is_R_or_C 𝕜] {n m : } (A B : matrix (fin n) (fin m) 𝕜) (i j : fin m) :

The inner product of a column of A and a column of B is an entry of Aᴴ ⬝ B