Complexification - Basic Properties

Basic Properties

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 + ib 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.

There is a natural embedding of V into VC given by

The vector space V may then be regarded as a real subspace of VC. If V has a basis {e_i} (over the field R) then a corresponding basis for VC is given by {e_i ⊗ 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 where J encodes the data of "multiplication by i". In matrix form, J 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, can be written as or identifying V 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.