Complexification - Linear Transformations

Linear Transformations

Given a real linear transformation f : V → W between two real vector spaces there is a natural complex linear transformation

given by

The map fC is naturally 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 fC commutes with conjugation and so maps the real subspace of VC to the real subspace of WC (via the map f). 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.