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| {{for|the complexification of a real Lie group|Complexification (Lie group)}}
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| In [[mathematics]], the '''complexification''' of a [[real vector space]] ''V'' is a vector space ''V''<sup>'''C'''</sup> over the [[complex number]] [[field (mathematics)|field]] obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any [[Basis (linear algebra)|basis]] for ''V'' over the real numbers serves as a basis for ''V''<sup>'''C'''</sup> over the complex numbers.
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| == Formal definition ==
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| 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):
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| :<math>V^{\mathbb C} = V\otimes_{\mathbb{R}} \mathbb{C}.</math>
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| 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). As it stands, ''V''<sup>'''C'''</sup> is only a real vector space. However, we can make ''V''<sup>'''C'''</sup> into a complex vector space by defining complex multiplication as follows:
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| :<math>\alpha(v\otimes \beta) = v\otimes(\alpha\beta)\qquad\mbox{for all } v\in V \mbox{ and }\alpha,\beta\in\mathbb C.</math>
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| 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.
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| Formally, complexification is a [[functor]] Vect<sub>'''R'''</sup> → Vect<sub>'''C'''</sup>, 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]] Vect<sub>'''C'''</sup> → Vect<sub>'''R'''</sup> from forgetting the complex structure.
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| == Basic properties ==
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| By the nature of the tensor product, every vector ''v'' in ''V''<sup>'''C'''</sup> can be written uniquely in the form
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| :<math>v = v_1\otimes 1 + v_2\otimes i</math>
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| where ''v''<sub>1</sub> and ''v''<sub>2</sub> are vectors in ''V''. It is a common practice to drop the tensor product symbol and just write
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| :<math>v = v_1 + iv_2.\,</math>
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| Multiplication by the complex number {{nowrap|''a'' + ''ib''}} is then given by the usual rule
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| :<math>(a+ib)(v_1 + iv_2) = (av_1 - bv_2) + i(bv_1 + av_2).\,</math>
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| We can then regard ''V''<sup>'''C'''</sup> as the [[direct sum of vector spaces|direct sum]] of two copies of ''V'':
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| :<math>V^{\mathbb C} \cong V\oplus iV</math> | |
| with the above rule for multiplication by complex numbers.
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| There is a natural embedding of ''V'' into ''V''<sup>'''C'''</sup> given by
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| :<math>v\mapsto v\otimes 1.</math>
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| The vector space ''V'' may then be regarded as a ''real'' [[linear subspace|subspace]] of ''V''<sup>'''C'''</sup>. If ''V'' has a [[basis (linear algebra)|basis]] {''e''<sub>''i''</sub>} (over the field '''R''') then a corresponding basis for ''V''<sup>'''C'''</sup> is given by {{nowrap|{''e''<sub>''i''</sub> ⊗ 1} }} over the field '''C'''. The complex [[dimension (linear algebra)|dimension]] of ''V''<sup>'''C'''</sup> is therefore equal to the real dimension of ''V'':
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| :<math>\dim_{\mathbb C}V^{\mathbb C} = \dim_{\mathbb R}V.</math>
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| Alternatively, rather than using tensor products, one can use this direct sum as the ''definition'' of the complexification:
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| :<math>V^{\mathbb C} := V \oplus V,</math>
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| where <math>V^{\mathbb C}</math> is given a [[linear complex structure]] by the operator ''J'' defined as <math>J(v,w) := (-w,v),</math> where ''J'' encodes the data of "multiplication by ''i''". In matrix form, ''J'' is given by:
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| :<math>J = \begin{bmatrix}0 & -I_V \\ I_V & 0\end{bmatrix}.</math>
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| 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^{\mathbb C}</math> can be written as <math>V \oplus JV</math> or <math>V \oplus iV,</math> identifying ''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.
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| == Examples ==
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| *The complexification of [[real coordinate space]] '''R'''<sup>''n''</sup> is complex coordinate space '''C'''<sup>''n''</sup>.
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| *Likewise, if ''V'' consists of the ''m''×''n'' [[matrix (mathematics)|matrices]] with real entries, ''V''<sup>'''C'''</sup> would consist of ''m''×''n'' matrices with complex entries.
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| *The complexification of [[quaternion]]s is the [[biquaternion]]s.
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| *The complexification of the [[split-complex number]]s is the [[tessarine]]s.
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| == Complex conjugation ==
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| The complexified vector space ''V''<sup>'''C'''</sup> has more structure than an ordinary complex vector space.{{examples needed|date=August 2013}} It comes with a [[canonical form|canonical]] [[complex conjugation]] map:
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| :<math>\chi : V^{\mathbb C} \to \overline{V^{\mathbb C}}</math>
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| defined by
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| :<math>\chi(v\otimes z) = v\otimes \bar z.</math>
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| The map χ may either be regarded as a [[conjugate-linear map]] from ''V''<sup>'''C'''</sup> to itself or as a complex linear [[isomorphism]] from ''V''<sup>'''C'''</sup> to its [[complex conjugate vector space|complex conjugate]] <math>\overline {V^{\mathbb C}}</math>.
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| Conversely, given a complex vector space ''W'' with a complex conjugation χ, ''W'' is isomorphic as a complex vector space to the complexification ''V''<sup>'''C'''</sup> of the real subspace
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| :<math>V = \{w\in W : \chi(w) = w\}.</math>
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| In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.
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| For example, when ''W'' = '''C'''<sup>''n''</sup> with the standard complex conjugation
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| :<math>\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n)</math>
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| the invariant subspace ''V'' is just the real subspace '''R'''<sup>''n''</sup>.
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| == Linear transformations ==
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| Given a real [[linear transformation]] ''f'' : ''V'' → ''W'' between two real vector spaces there is a natural complex linear transformation
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| :<math>f^{\mathbb C} : V^{\mathbb C} \to W^{\mathbb C}</math>
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| given by
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| :<math>f^{\mathbb C}(v\otimes z) = f(v)\otimes z.</math>
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| The map ''f''<sup>'''C'''</sup> is naturally called the '''complexification''' of ''f''. The complexification of linear transformations satisfies the following properties
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| *<math>(\mathrm{id}_V)^{\mathbb C} = \mathrm{id}_{V^{\mathbb C}}</math>
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| *<math>(f\circ g)^{\mathbb C} = f^{\mathbb C}\circ g^{\mathbb C}</math>
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| *<math>(f+g)^{\mathbb C} = f^{\mathbb C} + g^{\mathbb C}</math>
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| *<math>(af)^{\mathbb C} = af^{\mathbb C}\quad \forall a\in\mathbb R</math>
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| 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.
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| The map ''f''<sup>'''C'''</sup> commutes with conjugation and so maps the real subspace of ''V''<sup>'''C'''</sup> to the real subspace of ''W''<sup>'''C'''</sup> (via the map ''f''). Moreover, a complex linear map ''g'' : ''V''<sup>'''C'''</sup> → ''W''<sup>'''C'''</sup> is the complexification of a real linear map if and only if it commutes with conjugation.
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| As an example consider a linear transformation from '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup> thought of as an ''m'' × ''n'' [[matrix (mathematics)|matrix]]. The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from '''C'''<sup>''n''</sup> to '''C'''<sup>''m''</sup>.
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| == Dual spaces and tensor products ==
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| The [[dual space|dual]] of a real vector space ''V'' is the space ''V''* of all real linear maps from ''V'' to '''R'''. The complexification of ''V''* can naturally be thought of as the space of all real linear maps from ''V'' to '''C''' (denoted Hom<sub>'''R'''</sub>(''V'','''C''')). That is,
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| :<math>(V^*)^{\mathbb C} = V^*\otimes \mathbb C \cong \mathrm{Hom}_{\mathbb R}(V,\mathbb C).</math>
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| The isomorphism is given by
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| :<math>(\varphi_1\otimes 1 + \varphi_2\otimes i) \leftrightarrow \varphi_1 + i\varphi_2</math>
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| where φ<sub>1</sub> and φ<sub>2</sub> are elements of ''V''*. Complex conjugation is then given by the usual operation
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| :<math>\overline{\varphi_1 + i\varphi_2} = \varphi_1 - i\varphi_2</math>
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| Given a real linear map φ : ''V'' → '''C''' we may extend by linearity to obtain a complex linear map φ : ''V''<sup>'''C'''</sup> → '''C'''. That is,
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| :<math>\varphi(v\otimes z) = z\varphi(v).</math> | |
| This extension gives an isomorphism from Hom<sub>'''R'''</sub>(''V'','''C''')) to Hom<sub>'''C'''</sub>(''V''<sup>'''C'''</sup>,'''C'''). The latter is just the ''complex'' dual space to ''V''<sup>'''C'''</sup>, so we have a [[natural isomorphism]]:
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| :<math>(V^*)^{\mathbb C} \cong (V^{\mathbb C})^*.</math>
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| More generally, given real vector spaces ''V'' and ''W'' there is a natural isomorphism
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| :<math>\mathrm{Hom}_{\mathbb R}(V,W)^{\mathbb C} \cong \mathrm{Hom}_{\mathbb C}(V^{\mathbb C},W^{\mathbb C}).</math>
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| Complexification also commutes with the operations of taking [[tensor product]]s, [[exterior power]]s and [[symmetric power]]s. For example, if ''V'' and ''W'' are real vector spaces there is a natural isomorphism
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| :<math>(V\otimes_{\mathbb R}W)^{\mathbb C} \cong V^{\mathbb C}\otimes_{\mathbb C}W^{\mathbb C}.</math>
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| 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
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| :<math>(\Lambda_{\mathbb R}^k V)^{\mathbb C} \cong \Lambda_{\mathbb C}^k (V^{\mathbb C}).</math>
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| In all cases, the isomorphisms are the “obvious” ones.
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| == See also ==
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| *[[Extension of scalars]] – general process
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| *[[Linear complex structure]]
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| == References ==
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| * [[Paul Halmos]] (1958, 1974) ''Finite-Dimensional Vector Spaces'', p 41 and §77 Complexification, pp 150–153, Springer, ISBN 0-387-90093-4 .
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| * Ronald Shaw (1982) ''Linear Algebra and Group Representations'', v. 1, §1.5.4 Complexification and realification, pp 40–2 & §5.5.2 Complexification p 196, [[Academic Press]] ISBN 0-12-639201-3 .
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| *{{cite book | first = Steven | last = Roman | title = Advanced Linear Algebra | edition = (2nd ed.) | series = Graduate Texts in Mathematics '''135''' | publisher = Springer | location = New York | year = 2005 | isbn = 0-387-24766-1}}
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| [[Category:Complex manifolds]]
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| [[Category:Vector spaces]]
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