Complexification
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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
- <math>V^C=V\otimes_{\mathbb{R}} \mathbb{C}<math>,
the tensor product over the real field of V and the complex numbers.
<math>V^C<math> is a complex vector space with additional structure: a canonical complex conjugation <math>\phi<math>. Indeed, since <math>V<math> is included in <math>V^C<math> by <math>v\mapsto v\otimes 1<math>, the complex conjugation is defined by <math>\phi(v\otimes z) = v\otimes z^*<math>. This operation is usually denoted by <math>w^*<math> or <math>\overline{w}<math>.
Conversely, given a complex vector space <math>W<math> with a complex conjugation <math>\phi<math>, <math>W<math> is isomorphic as a complex vector space to the complexification <math>S^C<math> of the real subspace of <math>W<math>: <math> S = \{w\in W : \phi(w) = w\} <math>. In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.
For example, when <math>W=\mathbb{C}<math> with the standard complex conjugation <math>\phi(z) = z^*<math>, <math>S=\mathbb{R}<math>.