Editing Complexification

{{Short description|Topic in mathematics}}
{{for|the complexification of a real Lie group|Complexification (Lie group)}}
In [[mathematics]], the '''complexification''' of a [[vector space]] {{math|''V''}} over the field of real numbers (a "real vector space") yields a vector space {{math|''V''{{i sup|'''C'''}}}} over the [[complex number]] [[field (mathematics)|field]], obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any [[Basis (linear algebra)|basis]] for {{math|''V''}} (a space over the real numbers) may also serve as a basis for {{math|''V''{{i sup|'''C'''}}}} over the complex numbers.

== Formal definition ==
Let <math>V</math> be a real vector space. The '''{{em|{{visible anchor|complexification}}}}''' of {{math|''V''}} is defined by taking the [[tensor product]] of <math>V</math> with the complex numbers (thought of as a 2-dimensional vector space over the reals):

:<math>V^{\Complex} = V\otimes_{\R} \Complex\,.</math>

The subscript, <math>\R</math>, on the tensor product indicates that the tensor product is taken over the real numbers (since <math>V</math> is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, <math>V^{\Complex}</math> is only a real vector space. However, we can make <math>V^{\Complex}</math> into a complex vector space by defining complex multiplication as follows:

:<math>\alpha(v \otimes \beta) = v\otimes(\alpha\beta)\qquad\mbox{ for all } v\in V \mbox{ and }\alpha,\beta \in \Complex.</math>

More generally, complexification is an example of [[extension of scalars]] – here extending scalars from the real numbers to the complex numbers – which can be done for any [[field extension]], or indeed for any morphism of rings.

Formally, complexification is a [[functor]] {{math|Vect<sub>'''R'''</sub> → Vect<sub>'''C'''</sub>}}, from the category of real vector spaces to the category of complex vector spaces. This is the [[adjoint functor]] – specifically the [[left adjoint]] – to the [[forgetful functor]] {{math|Vect<sub>'''C'''</sub> → Vect<sub>'''R'''</sub>}} forgetting the complex structure.

This forgetting of the complex structure of a complex vector space <math>V</math> is called '''{{em|{{visible anchor|decomplexification}}}}''' (or sometimes "'''{{em|{{visible anchor|realification}}}}'''"). The decomplexification of a complex vector space <math>V</math> with basis <math>e_{\mu}</math> removes the possibility of complex multiplication of scalars, thus yielding a real vector space <math>W_{\R}</math> of twice the dimension with a basis <math>\{e_{\mu}, ie_{\mu}\}.</math><ref>{{cite book|last1=Kostrikin|first1=Alexei I.|last2=Manin|first2=Yu I.|title=Linear Algebra and Geometry|date=July 14, 1989|publisher=CRC Press|isbn=978-2881246838|page=75}}</ref>

== Basic properties ==
By the nature of the tensor product, every vector {{math|''v''}} in {{math|''V''{{i sup|'''C'''}}}} can be written uniquely in the form
:<math>v = v_1\otimes 1 + v_2\otimes i</math>
where {{math|''v''<sub>1</sub>}} and {{math|''v''<sub>2</sub>}} are vectors in {{math|''V''}}. It is a common practice to drop the tensor product symbol and just write
:<math>v = v_1 + iv_2.\,</math>
Multiplication by the complex number {{math|''a'' + ''i b''}} is then given by the usual rule
:<math>(a+ib)(v_1 + iv_2) = (av_1 - bv_2) + i(bv_1 + av_2).\,</math>
We can then regard {{math|''V''{{i sup|'''C'''}}}} as the [[direct sum of vector spaces|direct sum]] of two copies of {{math|''V''}}:
:<math>V^{\Complex} \cong V \oplus i V</math>
with the above rule for multiplication by complex numbers.

There is a natural embedding of {{math|''V''}} into {{math|''V''{{i sup|'''C'''}}}} given by
:<math>v\mapsto v\otimes 1.</math>
The vector space {{math|''V''}} may then be regarded as a ''real'' [[linear subspace|subspace]] of {{math|''V''{{i sup|'''C'''}}}}. If {{math|''V''}} has a [[basis (linear algebra)|basis]] {{math|{{mset| ''e''<sub>''i''</sub> }}}} (over the field {{math|'''R'''}}) then a corresponding basis for {{math|''V''{{i sup|'''C'''}}}} is given by {{math|{ ''e''<sub>''i''</sub> ⊗ 1 } }} over the field {{math|'''C'''}}. The complex [[dimension (linear algebra)|dimension]] of {{math|''V''{{i sup|'''C'''}}}} is therefore equal to the real dimension of {{math|''V''}}:

:<math>\dim_{\Complex} V^{\Complex} = \dim_{\R} V.</math>

Alternatively, rather than using tensor products, one can use this direct sum as the ''definition'' of the complexification:
:<math>V^{\Complex} := V \oplus V,</math>
where <math>V^{\Complex}</math> is given a [[linear complex structure]] by the operator {{math|''J''}} defined as <math>J(v,w) := (-w,v),</math> where {{math|''J''}} encodes the operation of “multiplication by {{mvar|i}}”. In matrix form, {{math|''J''}} is given by:
:<math>J = \begin{bmatrix}0 & -I_V \\ I_V & 0\end{bmatrix}.</math>
This yields the identical space – a real vector space with linear complex structure is identical data to a complex vector space – though it constructs the space differently. Accordingly, <math>V^{\Complex}</math> can be written as <math>V \oplus JV</math> or <math>V \oplus i V,</math> identifying {{math|''V''}} with the first direct summand. This approach is more concrete, and has the advantage of avoiding the use of the technically involved tensor product, but is ad hoc.

== Examples ==
* The complexification of [[real coordinate space]] {{math|'''R'''<sup>''n''</sup>}} is the complex coordinate space {{math|'''C'''<sup>''n''</sup>}}.
* Likewise, if {{math|''V''}} consists of the {{math|''m''×''n''}} [[matrix (mathematics)|matrices]] with real entries, {{math|''V''{{i sup|'''C'''}}}} would consist of {{math|''m''×''n''}} matrices with complex entries.

== Dickson doubling ==
{{Main|Cayley–Dickson construction}}
The process of complexification by moving from {{math|'''R'''}} to {{math|'''C'''}} was abstracted by twentieth-century mathematicians including [[Leonard Dickson]]. One starts with using the [[identity mapping]] {{math|1=''x''* = ''x''}} as a trivial [[involution (mathematics)|involution]] on {{math|'''R'''}}. Next two copies of '''R''' are used to form {{math|1=''z'' = (''a , b'')}} with the [[complex conjugation]] introduced as the involution {{math|1=''z''* = (''a'', −''b'')}}. Two elements {{mvar|w}} and {{mvar|z}} in the doubled set multiply by
:<math>w z = (a,b) \times (c,d) = (ac\ - \ d^*b,\ da \ + \ b c^*).</math>
Finally, the doubled set is given a '''norm''' {{math|1=''N''(''z'') = ''z* z''}}. When starting from {{math|'''R'''}} with the identity involution, the doubled set is {{math|'''C'''}} with the norm {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}}.
If one doubles {{math|'''C'''}}, and uses conjugation (''a,b'')* = (''a''*, –''b''), the construction yields [[quaternion]]s. Doubling again produces [[octonion]]s, also called Cayley numbers. It was at this point that Dickson in 1919 contributed to uncovering algebraic structure.

The process can also be initiated with {{math|'''C'''}} and the trivial involution {{math|1=''z''* = ''z''}}. The norm produced is simply {{math|''z''<sup>2</sup>}}, unlike the generation of {{math|'''C'''}} by doubling {{math|'''R'''}}. When this {{math|'''C'''}} is doubled it produces [[bicomplex number]]s, and doubling that produces [[biquaternion]]s, and doubling again results in [[bioctonion]]s. When the base algebra is associative, the algebra produced by this Cayley–Dickson construction is called a [[composition algebra]] since it can be shown that it has the property
:<math>N(p\,q) = N(p)\,N(q)\,.</math>

== Complex conjugation ==
The complexified vector space {{math|''V''{{i sup|'''C'''}}}} has more structure than an ordinary complex vector space. It comes with a [[canonical form|canonical]] [[complex conjugation]] map:
:<math>\chi : V^{\Complex} \to \overline{V^{\Complex}}</math>
defined by
:<math>\chi(v\otimes z) = v\otimes \bar z.</math> 
The map {{mvar|χ}} may either be regarded as a [[conjugate-linear map]] from {{math|''V''{{i sup|'''C'''}}}} to itself or as a complex linear [[isomorphism]] from {{math|''V''{{i sup|'''C'''}}}} to its [[complex conjugate vector space|complex conjugate]] <math>\overline {V^{\Complex}}</math>.

Conversely, given a complex vector space {{math|''W''}} with a complex conjugation {{mvar|χ}}, {{math|''W''}} is isomorphic as a complex vector space to the complexification {{math|''V''{{i sup|'''C'''}}}} of the real subspace
:<math>V = \{ w \in W : \chi(w) = w \}.</math>
In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.

For example, when {{math|1=''W'' = '''C'''<sup>''n''</sup>}} with the standard complex conjugation
:<math>\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n)</math>
the invariant subspace {{math|''V''}} is just the real subspace {{math|'''R'''<sup>''n''</sup>}}.

== Linear transformations ==
Given a real [[linear transformation]] {{math|''f'' : ''V'' → ''W''}}  between two real vector spaces there is a natural complex linear transformation
:<math>f^{\Complex} : V^{\Complex} \to W^{\Complex}</math>
given by
:<math>f^{\Complex}(v\otimes z) = f(v)\otimes z.</math>
The map <math>f^{\Complex}</math> is called the '''complexification''' of ''f''. The complexification of linear transformations satisfies the following properties
*<math>(\mathrm{id}_V)^{\Complex} = \mathrm{id}_{V^{\Complex}}</math>
*<math>(f \circ g)^{\Complex} = f^{\Complex} \circ g^{\Complex}</math>
*<math>(f+g)^{\Complex} = f^{\Complex} + g^{\Complex}</math>
*<math>(a f)^{\Complex} = a f^{\Complex} \quad \forall a \in \R</math>

In the language of [[category theory]] one says that complexification defines an ([[additive functor|additive]]) [[functor]] from the [[category of vector spaces|category of real vector spaces]] to the category of complex vector spaces.

The map {{math|''f''{{i sup|'''C'''}}}} commutes with conjugation and so maps the real subspace of ''V''{{i sup|'''C'''}} to the real subspace of {{math|''W''{{i sup|'''C'''}}}} (via the map {{math|''f''}}). Moreover, a complex linear map {{math|''g'' : ''V''{{i sup|'''C'''}} → ''W''{{i sup|'''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 {{math|'''R'''<sup>''n''</sup>}} to {{math|'''R'''<sup>''m''</sup>}} thought of as an {{math|''m''×''n''}} [[matrix (mathematics)|matrix]]. The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from {{math|'''C'''<sup>''n''</sup>}} to {{math|'''C'''<sup>''m''</sup>}}.

==Dual spaces and tensor products==

The [[dual space|dual]] of a real vector space {{math|''V''}} is the space {{math|''V''*}} of all real linear maps from {{math|''V''}} to {{math|'''R'''}}. The complexification of {{math|''V''*}} can naturally be thought of as the space of all real linear maps from {{math|''V''}} to {{math|'''C'''}} (denoted {{math|Hom<sub>'''R'''</sub>(''V'','''C''')}}). That is,
<math display=block>(V^*)^{\Complex} = V^*\otimes \Complex \cong \mathrm{Hom}_{\Reals}(V,\Complex).</math>

The isomorphism is given by
<math display=block>(\varphi_1\otimes 1 + \varphi_2\otimes i) \leftrightarrow \varphi_1 + i \varphi_2</math>
where {{math|''φ''<sub>1</sub>}} and {{math|''φ''<sub>2</sub>}} are elements of {{math|''V''*}}. Complex conjugation is then given by the usual operation
<math display=block>\overline{\varphi_1 + i\varphi_2} = \varphi_1 - i \varphi_2.</math>

Given a real linear map {{math|''φ'' : ''V'' → '''C'''}} we may [[extend by linearity]] to obtain a complex linear map {{math|''φ'' : ''V''{{i sup|'''C'''}} → '''C'''}}. That is,
<math display=block>\varphi(v\otimes z) = z\varphi(v).</math>
This extension gives an isomorphism from {{math|Hom<sub>'''R'''</sub>(''V'','''C''')}} to {{math|Hom<sub>'''C'''</sub>(''V''{{i sup|'''C'''}},'''C''')}}. The latter is just the ''complex'' dual space to {{math|''V''{{i sup|'''C'''}}}}, so we have a [[natural isomorphism]]:
<math display=block>(V^*)^{\Complex} \cong (V^{\Complex})^*.</math>

More generally, given real vector spaces {{math|''V''}} and {{math|''W''}} there is a natural isomorphism
<math display=block>\mathrm{Hom}_{\Reals}(V,W)^{\Complex} \cong \mathrm{Hom}_{\Complex}(V^{\Complex},W^{\Complex}).</math>

Complexification also commutes with the operations of taking [[tensor product]]s, [[exterior power]]s and [[symmetric power]]s. For example, if {{math|''V''}} and {{math|''W''}} are real vector spaces there is a natural isomorphism
<math display=block>(V \otimes_{\Reals} W)^{\Complex} \cong V^{\Complex} \otimes_{\Complex} W^{\Complex}\,.</math>
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
<math display=block>(\Lambda_{\Reals}^k V)^{\Complex} \cong \Lambda_{\Complex}^k (V^{\Complex}).</math>
In all cases, the isomorphisms are the “obvious” ones.

== See also ==
*[[Extension of scalars]] – general process
*[[Linear complex structure]]
*[[Baker–Campbell–Hausdorff formula]]

== References ==
{{reflist}}

* {{cite book |first=Paul |last=Halmos |author-link=Paul Halmos |orig-year=1958 |year=1974 |title=Finite-Dimensional Vector Spaces |at=p&nbsp;41 and §77&nbsp;Complexification, pp&nbsp;150–153 |publisher=Springer |ISBN=0-387-90093-4}}
*{{cite book |last=Shaw |first=Ronald |title=Linear Algebra and Group Representations |volume=I: Linear Algebra and Introduction to Group Representations |year=1982 |publisher=Academic Press |isbn=0-12-639201-3 |page=[https://archive.org/details/linearalgebragro0000shaw/page/196 196] |url=https://archive.org/details/linearalgebragro0000shaw/page/196}}
*{{cite book |first=Steven |last=Roman |title=Advanced Linear Algebra |edition=2nd |series=Graduate Texts in Mathematics |volume=135 |publisher=Springer |location=New York |year=2005 |isbn=0-387-24766-1}}

[[Category:Complex manifolds]]
[[Category:Vector spaces]]