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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the complexification of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over the field of real numbers (a "real vector space") yields a vector space over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
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.


Formal definition

Let V be a real vector space. The of is defined by taking the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): :V^ = V\otimes_ \Complex\,. 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^ is only a real vector space. However, we can make V^ into a complex vector space by defining complex multiplication as follows: :\alpha(v \otimes \beta) = v\otimes(\alpha\beta)\qquad\mbox v\in V \mbox\alpha,\beta \in \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 In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
, or indeed for any morphism of rings. Formally, complexification is a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
, from the category of real vector spaces to the category of complex vector spaces. This is the
adjoint functor In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
– specifically the
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...
– to the
forgetful functor In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of ...
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_ removes the possibility of complex multiplication of scalars, thus yielding a real vector space W_ of twice the dimension with a basis \.


Basic properties

By the nature of the tensor product, every vector in can be written uniquely in the form :v = v_1\otimes 1 + v_2\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.\, 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).\, We can then regard as the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of two copies of : :V^ \cong V \oplus i V with the above rule for multiplication by complex numbers. There is a natural embedding of into given by :v\mapsto v\otimes 1. The vector space may then be regarded as a ''real'' subspace of . If has a basis (over the field ) then a corresponding basis for is given by over the field . The complex
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of is therefore equal to the real dimension of : :\dim_ V^ = \dim_ V. Alternatively, rather than using tensor products, one can use this direct sum as the ''definition'' of the complexification: :V^ := V \oplus V, where V^ is given a
linear complex structure In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, - \text_V . Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to re ...
by the operator defined as J(v,w) := (-w,v), where encodes the operation of “multiplication by ”. In matrix form, is given by: :J = \begin0 & -I_V \\ I_V & 0\end. 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^ can be written as V \oplus JV or V \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.


Examples

* The complexification of
real coordinate space In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as '' coordinate vectors''. ...
is the complex coordinate space . * Likewise, if consists of the
matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...
with real entries, would consist of matrices with complex entries.


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 Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
as a trivial
involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...
on . Next two copies of R are used to form with the
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
introduced as the involution . Two elements and in the doubled set multiply by :w z = (a,b) \times (c,d) = (ac\ - \ d^*b,\ da \ + \ 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
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s. Doubling again produces
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...
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 and the trivial involution . The norm produced is simply , unlike the generation of by doubling . When this is doubled it produces bicomplex numbers, and doubling that produces
biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions cor ...
s, and doubling again results in bioctonions. When the base algebra is associative, the algebra produced by this Cayley–Dickson construction is called a
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution ...
since it can be shown that it has the property :N(p\,q) = N(p)\,N(q)\,.


Complex conjugation

The complexified vector space has more structure than an ordinary complex vector space. It comes with a
canonical The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical exampl ...
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
map: :\chi : V^ \to \overline defined by :\chi(v\otimes z) = v\otimes \bar z. The map may either be regarded as a conjugate-linear map from to itself or as a complex linear
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
from to its
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
\overline . 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 = \. 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 :\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n) the invariant subspace is just the real subspace .


Linear transformations

Given a real
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
between two real vector spaces there is a natural complex linear transformation :f^ : V^ \to W^ given by :f^(v\otimes z) = f(v)\otimes z. The map f^ is called the complexification of ''f''. The complexification of linear transformations satisfies the following properties *(\mathrm_V)^ = \mathrm_ *(f \circ g)^ = f^ \circ g^ *(f+g)^ = f^ + g^ *(a f)^ = a f^ \quad \forall a \in \R In the language of
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
one says that complexification defines an (
additive Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-function see Sigma additivity * Additive category, a preadditive category with fin ...
)
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
. The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from to .


Dual spaces and tensor products

The dual 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^*)^ = V^*\otimes \Complex \cong \mathrm_(V,\Complex). The isomorphism is given by (\varphi_1\otimes 1 + \varphi_2\otimes i) \leftrightarrow \varphi_1 + i \varphi_2 where and are elements of . Complex conjugation is then given by the usual operation \overline = \varphi_1 - i \varphi_2. Given a real linear map we may
extend by linearity In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
to obtain a complex linear map . That is, \varphi(v\otimes z) = z\varphi(v). This extension gives an isomorphism from to . The latter is just the ''complex'' dual space to , so we have a
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
: (V^*)^ \cong (V^)^*. More generally, given real vector spaces and there is a natural isomorphism \mathrm_(V,W)^ \cong \mathrm_(V^,W^). Complexification also commutes with the operations of taking
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
s,
exterior power In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
s and
symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n. More precisely, the notion exists at least in th ...
s. For example, if and are real vector spaces there is a natural isomorphism (V \otimes_ W)^ \cong V^ \otimes_ W^\,. 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_^k V)^ \cong \Lambda_^k (V^). In all cases, the isomorphisms are the “obvious” ones.


See also

* Extension of scalars – general process *
Linear complex structure In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, - \text_V . Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to re ...
*
Baker–Campbell–Hausdorff formula In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation e^X e^Y = e^Z for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultima ...


References

* * *{{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 Complex manifolds Vector spaces