mathlib documentation

data.complex.module

Complex number as a vector space over #

This file contains the following instances:

It also defines bundled versions of four standard maps (respectively, the real part, the imaginary part, the embedding of in , and the complex conjugate):

It also provides a universal property of the complex numbers complex.lift, which constructs a ℂ →ₐ[ℝ] A into any -algebra A given a square root of -1.

In addition, this file provides a decomposition into real_part and imaginary_part for any element of a star_module over .

Notation #

@[protected, instance]
def complex.has_smul {R : Type u_1} [has_smul R ] :
Equations
theorem complex.smul_re {R : Type u_1} [has_smul R ] (r : R) (z : ) :
(r z).re = r z.re
theorem complex.smul_im {R : Type u_1} [has_smul R ] (r : R) (z : ) :
(r z).im = r z.im
@[simp]
theorem complex.real_smul {x : } {z : } :
x z = x * z
@[protected, instance]
@[protected, instance]
@[protected, instance]
@[simp]
theorem alg_hom.map_coe_real_complex {A : Type u_3} [semiring A] [algebra A] (f : →ₐ[] A) (x : ) :

We need this lemma since complex.coe_algebra_map diverts the simp-normal form away from alg_hom.commutes.

@[ext]
theorem complex.alg_hom_ext {A : Type u_3} [semiring A] [algebra A] ⦃f g : →ₐ[] A⦄ (h : f complex.I = g complex.I) :
f = g

Two -algebra homomorphisms from ℂ are equal if they agree on complex.I.

@[protected]
noncomputable def complex.basis_one_I  :

has a basis over given by 1 and I.

Equations

fact version of the dimension of over , locally useful in the definition of the circle.

@[protected]
noncomputable def complex.orientation  :

The standard orientation on .

Equations
@[protected, instance]
Equations
@[protected, instance]
@[simp, norm_cast]
theorem complex.coe_smul {E : Type u_1} [add_comm_group E] [module E] (x : ) (y : E) :
x y = x y
@[protected, instance]

The scalar action of on a -module E induced by module.complex_to_real commutes with another scalar action of M on E whenever the action of commutes with the action of M.

@[protected, instance]

Linear map version of the real part function, from to .

Equations

Linear map version of the imaginary part function, from to .

Equations

-algebra morphism version of the canonical embedding of in .

Equations

-algebra isomorphism version of the complex conjugation function from to

Equations

The identity and the complex conjugation are the only two -algebra homomorphisms of .

def complex.lift_aux {A : Type u_1} [ring A] [algebra A] (I' : A) (hf : I' * I' = -1) :

There is an alg_hom from to any -algebra with an element that squares to -1.

See complex.lift for this as an equiv.

Equations
@[simp]
theorem complex.lift_aux_apply {A : Type u_1} [ring A] [algebra A] (I' : A) (hI' : I' * I' = -1) (z : ) :
theorem complex.lift_aux_apply_I {A : Type u_1} [ring A] [algebra A] (I' : A) (hI' : I' * I' = -1) :
def complex.lift {A : Type u_1} [ring A] [algebra A] :
{I' // I' * I' = -1} ( →ₐ[] A)

A universal property of the complex numbers, providing a unique ℂ →ₐ[ℝ] A for every element of A which squares to -1.

This can be used to embed the complex numbers in the quaternions.

This isomorphism is named to match the very similar zsqrtd.lift.

Equations
@[simp]
theorem complex.lift_apply {A : Type u_1} [ring A] [algebra A] (I' : {I' // I' * I' = -1}) :

Create a self_adjoint element from a skew_adjoint element by multiplying by the scalar -complex.I.

Equations
noncomputable def real_part {A : Type u_1} [add_comm_group A] [module A] [star_add_monoid A] [star_module A] :

The real part ℜ a of an element a of a star module over , as a linear map. This is just self_adjoint_part, but we provide it as a separate definition in order to link it with lemmas concerning the imaginary_part, which doesn't exist in star modules over other rings.

Equations
noncomputable def imaginary_part {A : Type u_1} [add_comm_group A] [module A] [star_add_monoid A] [star_module A] :

The imaginary part ℑ a of an element a of a star module over , as a linear map into the self adjoint elements. In a general star module, we have a decomposition into the self_adjoint and skew_adjoint parts, but in a star module over we have real_part_add_I_smul_imaginary_part, which allows us to decompose into a linear combination of self_adjoints.

Equations
@[simp]

The standard decomposition of ℜ a + complex.I • ℑ a = a of an element of a star module over into a linear combination of self adjoint elements.