Complexification

In mathematics, the complexification of a real vector space V is a vector space V^C over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Any basis for V over the real numbers serves as a basis for V^C over the complex numbers. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

1 Formal definition
2 Basic properties
3 Examples
4 Complex conjugation
5 Linear transformations
6 Dual spaces and tensor products
7 See also
8 References

Formal definition

Let V be a real vector space. The complexification of V is defined by taking the tensor product of V with the complex numbers (thought of as a two-dimensional vector space over the reals):

$V^{\mathbb C} = V\otimes_{\mathbb{R}} \mathbb{C}.$

The subscript R on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector As it stands V^C is only a real vector space. However, we can make V^C into a complex vector space by defining complex multiplication as follows:

$\alpha(v\otimes \beta) = v\otimes(\alpha\beta)\qquad\mbox{for all } v\in V \mbox{ and }\alpha,\beta\in\mathbb C.$

Basic properties

By the nature of the tensor product, every vector v in V^C can be written uniquely in the form

$v = v_1\otimes 1 + v_2\otimes i$

where v₁ and v₂ are vectors in V. It is a common practice to drop the tensor product symbol and just write

$v = v_1 + iv_2.\,$

Multiplication by the complex number a + ib is then given by the usual rule

$(a+ib)(v_1 + iv_2) = (av_1 - bv_2) + i(bv_1 + av_2).\,$

We can then regard V^C as the direct sum of two copies of V:

$V^{\mathbb C} \cong V\oplus iV$

with the above rule for multiplication by complex numbers. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

There is a natural embedding of V into V^C given by

$v\mapsto v\otimes 1.$

The vector space V may then be regarded as a real subspace of V^C. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. If V has a basis {e_i} then a corresponding basis for V^C is given by {e_i⊗1}. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The complex dimension of V^C is therefore equal to the real dimension of V:

$\dim_{\mathbb C}V^{\mathbb C} = \dim_{\mathbb R}V.$

Examples

The complexification of real coordinate space Rⁿ is complex coordinate space Cⁿ. In Mathematics, the dimension of a Vector space V is the cardinality (i
Likewise, if V consists of the m×n matrices with real entries, V^C would consist of m×n matrices with complex entries. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

Complex conjugation

The complexified vector space V^C has more structure than an ordinary complex vector space. It comes with a canonical complex conjugation map:

$\chi : V^{\mathbb C} \to \overline{V^{\mathbb C}}$

defined by

$\chi(v\otimes z) = v\otimes \bar z.$

The map χ may either be regarded as a conjugate-linear map from V^C to itself or as a complex linear isomorphism from V^C to its complex conjugate $\overline {V^{\mathbb C}}$ . Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. In Mathematics, a mapping f: V → W from a Complex vector space to another is said to be antilinear (or conjugate-linear In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, one associates to every complex vector space V\ its complex conjugate vector space \overline V again a complex vector space

Conversely, given a complex vector space W with a complex conjugation χ, W is isomorphic as a complex vector space to the complexification V^C of the real subspace

$V = \{w\in W : \chi(w) = w\}.$

In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.

For example, when W = Cⁿ with the standard complex conjugation

$\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n)$

the invariant subspace V is just the real subspace Rⁿ.

Linear transformations

Given a real linear transformation f : V → W between two real vector spaces there is a natural complex linear transformation

$f^{\mathbb C} : V^{\mathbb C} \to W^{\mathbb C}$

given by

$f^{\mathbb C}(v\otimes z) = f(v)\otimes z.$

The map f^C is naturally called the complexification of f. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The complexification of linear transformations satisfies the following properties

$(\mathrm{id}_V)^{\mathbb C} = \mathrm{id}_{V^{\mathbb C}}$
$(f\circ g)^{\mathbb C} = f^{\mathbb C}\circ g^{\mathbb C}$
$(f+g)^{\mathbb C} = f^{\mathbb C} + g^{\mathbb C}$
$(af)^{\mathbb C} = af^{\mathbb C}\quad \forall a\in\mathbb R$

In the language of category theory one says that complexification defines an (additive) functor from the category of real vector spaces to the category of complex vector spaces. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects

The map f^C commutes with conjugation and so maps the real subspace of V^C to the real subspace of W^C (via the map f). Moreover, a complex linear map g : V^C → W^C is the complexification of a real linear map if and only if it commutes with conjugation.

As an example consider a linear transformation from Rⁿ to R^m thought of as an m × n matrix. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The complexification of that transformation is the exact same matrix, but now thought of as a linear map from Cⁿ to C^m.

Dual spaces and tensor products

The dual of a real vector space V is the space V* of all real linear maps from V to R. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals The complexification of V* can naturally be thought of as the space of all real linear maps from V to C (denoted Hom_R(V,C)). That is,

$(V^*)^{\mathbb C} = V^*\otimes \mathbb C \cong \mathrm{Hom}_{\mathbb R}(V,\mathbb C).$

The isomorphism is given by

$(\varphi_1\otimes 1 + \varphi_2\otimes i) \leftrightarrow \varphi_1 + i\varphi_2$

where φ₁ and φ₂ are elements of V*. Complex conjugation is then given by the usual operation

$\overline{\varphi_1 + i\varphi_2} = \varphi_1 - i\varphi_2$

Given a real linear map φ : V → C we may extend by linearity to obtain a complex linear map φ : V^C → C. That is,

$\varphi(v\otimes z) = z\varphi(v).$

This extension gives an isomorphism from Hom_R(V,C)) to Hom_C(V^C,C). The latter is just the complex dual space to V^C, so we have a natural isomorphism:

$(V^*)^{\mathbb C} \cong (V^{\mathbb C})^*.$

More generally, given real vector spaces V and W there is a natural isomorphism

$\mathrm{Hom}_{\mathbb R}(V,W)^{\mathbb C} \cong \mathrm{Hom}_{\mathbb C}(V^{\mathbb C},W^{\mathbb C}).$

Complexification also commutes with the operations of taking tensor products, exterior powers and symmetric powers. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field For example, if V and W are real vector spaces there is a natural isomorphism

$(V\otimes_{\mathbb R}W)^{\mathbb C} \cong V^{\mathbb C}\otimes_{\mathbb C}W^{\mathbb C}.$

Note the left-hand tensor product is taken over the reals while the right-hand one is taken over the complexes. The same pattern is true in general. For instance, one has

$(\Lambda_{\mathbb R}^k V)^{\mathbb C} \cong \Lambda_{\mathbb C}^k (V^{\mathbb C}).$

In all cases, the isomorphism are the “obvious” ones.

References

Roman, Steven (2005). In Mathematics, a complex structure on a Real vector space V is an Automorphism of V which squares to the minus identity Advanced Linear Algebra, (2nd ed. ), Graduate Texts in Mathematics 135, New York: Springer. ISBN 0-387-24766-1.

Dictionary

-noun

(uncountable, jocular) The act or process of making something more complex.
(countable, mathematics) An extension from a basis on real numbers to a basis on complex numbers.

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