Complexification

 ( Complex Manifolds ) 

In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers.

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Formal definition Let $V$ be a real vector space. The of is defined by taking the tensor product of $V$ with the complex numbers (thought of as a 2-dimensional vector space over the reals):

$V^\{\backslash Complex\}\; =\; V\backslash otimes\_\{\backslash R\}\; \backslash Complex\backslash ,.$

The subscript, $\backslash R$, 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, $V^\{\backslash Complex\}$ is only a real vector space. However, we can make $V^\{\backslash Complex\}$ into a complex vector space by defining complex multiplication as follows:

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

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 , 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 forgetting the complex structure.

This forgetting of the complex structure of a complex vector space $V$ is called ' (or sometimes "'"). The decomplexification of a complex vector space $V$ with basis $e\_\{\backslash mu\}$ removes the possibility of complex multiplication of scalars, thus yielding a real vector space $W\_\{\backslash R\}$ of twice the dimension with a basis $\backslash \{e\_\{\backslash mu\},\; ie\_\{\backslash mu\}\backslash \}.$

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Basic properties By the nature of the tensor product, every vector in can be written uniquely in the form

$v\; =\; v\_1\backslash otimes\; 1\; +\; v\_2\backslash otimes\; i$

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

$v\; =\; v\_1\; +\; iv\_2.\backslash ,$

Multiplication by the complex number is then given by the usual rule

$(a+ib)(v\_1\; +\; iv\_2)\; =\; (av\_1\; -\; bv\_2)\; +\; i(bv\_1\; +\; av\_2).\backslash ,$

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

$V^\{\backslash Complex\}\; \backslash cong\; V\; \backslash oplus\; i\; V$

with the above rule for multiplication by complex numbers.

There is a natural embedding of into given by

$v\backslash mapsto\; v\backslash otimes\; 1.$

The vector space may then be regarded as a real linear subspace of . If has a basis (over the field ) then a corresponding basis for is given by over the field . The complex dimension of is therefore equal to the real dimension of :

$\backslash dim\_\{\backslash Complex\}\; V^\{\backslash Complex\}\; =\; \backslash dim\_\{\backslash R\}\; V.$

Alternatively, rather than using tensor products, one can use this direct sum as the definition of the complexification:

$V^\{\backslash Complex\}\; :=\; V\; \backslash oplus\; V,$

where $V^\{\backslash Complex\}$ is given a linear complex structure by the operator defined as $J(v,w)\; :=\; (-w,v),$ where encodes the operation of “multiplication by ”. In matrix form, is given by:

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, $V^\{\backslash Complex\}$ can be written as $V\; \backslash oplus\; JV$ or $V\; \backslash oplus\; i\; V,$ identifying 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

• The complexification of real coordinate space is the complex coordinate space .
• Likewise, if consists of the matrices with real entries, would consist of matrices with complex entries.

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Dickson doubling The process of complexification by moving from to was abstracted by twentieth-century mathematicians including Leonard Dickson. One starts with using the identity mapping as a trivial involution on . Next two copies of R are used to form with the complex conjugation introduced as the involution . Two elements and in the doubled set multiply by

$w\; z\; =\; (a,b)\; \backslash times\; (c,d)\; =\; (ac\backslash \; -\; \backslash \; d^*b,\backslash \; da\; \backslash \; +\; \backslash \; b\; c^*).$

Finally, the doubled set is given a norm . When starting from with the identity involution, the doubled set is with the norm . If one doubles , and uses conjugation ( a,b)* = ( a*, – b), the construction yields . Doubling again produces , 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 and the trivial involution . The norm produced is simply , unlike the generation of by doubling . When this is doubled it produces , and doubling that produces , and doubling again results in . 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

$N(p\backslash ,q)\; =\; N(p)\backslash ,N(q)\backslash ,.$

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

$\backslash chi\; :\; V^\{\backslash Complex\}\; \backslash to\; \backslash overline\{V^\{\backslash Complex\}\}$

defined by

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

The map may either be regarded as a conjugate-linear map from to itself or as a complex linear isomorphism from to its complex conjugate $\backslash overline\; \{V^\{\backslash Complex\}\}$.

Conversely, given a complex vector space with a complex conjugation , is isomorphic as a complex vector space to the complexification of the real subspace

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

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

For example, when with the standard complex conjugation

$\backslash chi(z\_1,\backslash ldots,z\_n)\; =\; (\backslash bar\; z\_1,\backslash ldots,\backslash bar\; z\_n)$

the invariant subspace is just the real subspace .

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Linear transformations Given a real linear transformation between two real vector spaces there is a natural complex linear transformation

$f^\{\backslash Complex\}\; :\; V^\{\backslash Complex\}\; \backslash to\; W^\{\backslash Complex\}$

given by

$f^\{\backslash Complex\}(v\backslash otimes\; z)\; =\; f(v)\backslash otimes\; z.$

The map $f^\{\backslash Complex\}$ is called the complexification of f. The complexification of linear transformations satisfies the following properties

• $(\backslash mathrm\{id\}\_V)^\{\backslash Complex\}\; =\; \backslash mathrm\{id\}\_\{V^\{\backslash Complex\}\}$
• $(f\; \backslash circ\; g)^\{\backslash Complex\}\; =\; f^\{\backslash Complex\}\; \backslash circ\; g^\{\backslash Complex\}$
• $(f+g)^\{\backslash Complex\}\; =\; f^\{\backslash Complex\}\; +\; g^\{\backslash Complex\}$
• $(a\; f)^\{\backslash Complex\}\; =\; a\; f^\{\backslash Complex\}\; \backslash quad\; \backslash forall\; a\; \backslash in\; \backslash R$

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

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

As an example consider a linear transformation from to thought of as an matrix. The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from to .

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Dual spaces and tensor products The dual space of a real vector space is the space of all real linear maps from to . The complexification of can naturally be thought of as the space of all real linear maps from to (denoted ). That is, $$(V^*)^\{\backslash Complex\}\; =\; V^*\backslash otimes\; \backslash Complex\; \backslash cong\; \backslash mathrm\{Hom\}\_\{\backslash Reals\}(V,\backslash Complex).$$

The isomorphism is given by $$(\backslash varphi\_1\backslash otimes\; 1\; +\; \backslash varphi\_2\backslash otimes\; i)\; \backslash leftrightarrow\; \backslash varphi\_1\; +\; i\; \backslash varphi\_2$$ where and are elements of . Complex conjugation is then given by the usual operation $$\backslash overline\{\backslash varphi\_1\; +\; i\backslash varphi\_2\}\; =\; \backslash varphi\_1\; -\; i\; \backslash varphi\_2.$$

Given a real linear map we may extend by linearity to obtain a complex linear map . That is, $$\backslash varphi(v\backslash otimes\; z)\; =\; z\backslash varphi(v).$$ This extension gives an isomorphism from to . The latter is just the complex dual space to , so we have a natural isomorphism: $$(V^*)^\{\backslash Complex\}\; \backslash cong\; (V^\{\backslash Complex\})^*.$$

More generally, given real vector spaces and there is a natural isomorphism $$\backslash mathrm\{Hom\}\_\{\backslash Reals\}(V,W)^\{\backslash Complex\}\; \backslash cong\; \backslash mathrm\{Hom\}\_\{\backslash Complex\}(V^\{\backslash Complex\},W^\{\backslash Complex\}).$$

Complexification also commutes with the operations of taking , and . For example, if and are real vector spaces there is a natural isomorphism $$(V\; \backslash otimes\_\{\backslash Reals\}\; W)^\{\backslash Complex\}\; \backslash cong\; V^\{\backslash Complex\}\; \backslash otimes\_\{\backslash Complex\}\; W^\{\backslash Complex\}\backslash ,.$$ 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 $$(\backslash Lambda\_\{\backslash Reals\}^k\; V)^\{\backslash Complex\}\; \backslash cong\; \backslash Lambda\_\{\backslash Complex\}^k\; (V^\{\backslash Complex\}).$$ In all cases, the isomorphisms are the “obvious” ones.

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See also

• Extension of scalars – general process
• Linear complex structure
• Baker–Campbell–Hausdorff formula

• Paul Halmos (1974). 9780387900933, Springer. ISBN 9780387900933
• Ronald Shaw (1982). 9780126392012, Academic Press. . ISBN 9780126392012
• Steven Roman (2025). 9780387247663, Springer. ISBN 9780387247663

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