In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space VC over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers.
The complexification of V is defined by taking the tensor product of
with the complex numbers (thought of as a 2-dimensional vector space over the reals): The subscript,
is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted).
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 VectR → VectC, 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 VectC → VectR forgetting the complex structure.
removes the possibility of complex multiplication of scalars, thus yielding a real vector space
[1] By the nature of the tensor product, every vector v in VC can be written uniquely in the form where v1 and v2 are vectors in V. It is a common practice to drop the tensor product symbol and just write Multiplication by the complex number a + i b is then given by the usual rule We can then regard VC as the direct sum of two copies of V: with the above rule for multiplication by complex numbers.
If V has a basis { ei } (over the field R) then a corresponding basis for VC is given by { ei ⊗ 1 } over the field C. The complex dimension of VC is therefore equal to the real dimension of V: Alternatively, rather than using tensor products, one can use this direct sum as the definition of the complexification: where
is given a linear complex structure by the operator J defined as
This approach is more concrete, and has the advantage of avoiding the use of the technically involved tensor product, but is ad hoc.
The process of complexification by moving from R to C was abstracted by twentieth-century mathematicians including Leonard Dickson.
When starting from R with the identity involution, the doubled set is C with the norm a2 + b2.
If one doubles C, 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.
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 The complexified vector space VC has more structure than an ordinary complex vector space.
It comes with a canonical complex conjugation map: defined by The map χ may either be regarded as a conjugate-linear map from VC to itself or as a complex linear isomorphism from VC to its complex conjugate
Given a real linear transformation f : V → W between two real vector spaces there is a natural complex linear transformation given by The map
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.
Moreover, a complex linear map g : VC → WC is the complexification of a real linear map if and only if it commutes with conjugation.
As an example consider a linear transformation from Rn to Rm thought of as an m×n matrix.
The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from Cn to Cm.
The 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 HomR(V,C)).
This extension gives an isomorphism from HomR(V,C) to HomC(VC,C).
The latter is just the complex dual space to VC, so we have a natural isomorphism:
More generally, given real vector spaces V and W there is a natural isomorphism
For example, if V and W 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.