Complexification

In mathematics, the complexification of a vector space V over the real number field is the 'corresponding' vector space VC over the complex number field. That is, it shares the same dimension as V; and a basis for V over the real numbers can serve as a basis for VC over the complex numbers. For example, if V consists of the m×n matrices with real coefficients, VC would consist of m×n complex matrices. For the sake of having a basis-free definition, one can take
V^C=V\otimes_{\mathbb{R}} \mathbb{C},
the tensor product over the real field of V and the complex numbers.

 

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