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Concept
Complexification
Summary
In mathematics, the complexification of a vector space V over the field of real numbers (a "real vector space") yields a vector space V^C 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 V (a space over the real numbers) may also serve as a basis for V^C over the complex numbers.
Let be a real vector space. The 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, , 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 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.
This forgetting of the complex structure of a complex vector space is called (or sometimes ""). The decomplexification of a complex vector space with basis removes the possibility of complex multiplication of scalars, thus yielding a real vector space of twice the dimension with a basis
By the nature of the tensor product, every vector v in V^C 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 V^C as the direct sum of two copies of V:
with the above rule for multiplication by complex numbers.
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