Complexification

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.
Formal definition.
Let be a real vector space. The complexification of is defined by taking the tensor product of with the complex numbers (thought of as a 2dim(V)-dimensional vector space over the reals):
The subscript, formula_2, on the tensor product indicates that the tensor product is taken over the real numbers (since is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, is only a real vector space. However, we can make into a complex vector space by defining complex multiplication as follows:
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 Vect → Vect, 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 → Vect forgetting the complex structure.
This forgetting of the complex structure of a complex vector space formula_4 is called decomplexification (or sometimes "realification"). The decomplexification of a complex vector space formula_4 with basis formula_6 removes the possibility of complex multiplication of scalars, thus yielding a real vector space formula_7 of twice the dimension with a basis formula_8.
Basic properties.
By the nature of the tensor product, every vector in can be written uniquely in the form
where and are vectors in . It is a common practice to drop the tensor product symbol and just write
Multiplication by the complex number is then given by the usual rule
We can then regard as the direct sum of two copies of :
with the above rule for multiplication by complex numbers.
There is a natural embedding of into given by
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 of is therefore equal to the real dimension of :
Alternatively, rather than using tensor products, one can use this direct sum as the "definition" of the complexification:
where formula_16 is given a linear complex structure by the operator defined as formula_17 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, formula_19 can be written as formula_20 or formula_21 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.
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
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 quaternions. Doubling again produces octonions, 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 biquaternions, 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 since it can be shown that it has the property
Complex conjugation.
The complexified vector space has more structure than an ordinary complex vector space. It comes with a canonical complex conjugation map:
defined by
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 formula_26.
Conversely, given a complex vector space with a complex conjugation , is isomorphic as a complex vector space to the complexification of the real subspace
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
the invariant subspace is just the real subspace .
Linear transformations.
Given a real linear transformation between two real vector spaces there is a natural complex linear transformation
given by
The map formula_31 is called the complexification of "f". The complexification of linear transformations satisfies the following properties
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.
The map commutes with conjugation and so maps the real subspace of "V" ℂ 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 .
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,
The isomorphism is given by
where and are elements of . Complex conjugation is then given by the usual operation
Given a real linear map we may extend by linearity to obtain a complex linear map . That is,
This extension gives an isomorphism from to . The latter is just the "complex" dual space to , so we have a natural isomorphism:
More generally, given real vector spaces and there is a natural isomorphism
Complexification also commutes with the operations of taking tensor products, exterior powers and symmetric powers. For example, if and are real vector spaces there is a natural isomorphism
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
In all cases, the isomorphisms are the “obvious” ones.