Complexification

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

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^C is only a real vector space. However, we can make V^C 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_R → Vect_C, 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_C → Vect_R from forgetting the complex structure.

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

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