Complexification

Basic properties

By the nature of the tensor product, every vector v in V^C can be written uniquely in the form

${\displaystyle v=v_1\otimes 1+v_2\otimes i}$

where v₁ and v₂ are vectors in V. It is a common practice to drop the tensor product symbol and just write

${\displaystyle v=v_1+i{v}_2.}$

Multiplication by the complex number a + i b is then given by the usual rule

${\displaystyle (a+ib)(v_1+i{v}_2)=(a{v}_1-b{v}_2)+i(b{v}_1+a{v}_2).}$

We can then regard V^C as the direct sum of two copies of V:

${\displaystyle V^{\mathbb{C}}\cong V\oplus iV}$

with the above rule for multiplication by complex numbers.

There is a natural embedding of V into V^C given by

${\displaystyle v\mapsto v\otimes 1.}$

The vector space V may then be regarded as a real subspace of V^C. If V has a basis { e_i } (over the field R) then a corresponding basis for V^C is given by { e_i ⊗ 1 } over the field C. The complex dimension of V^C is therefore equal to the real dimension of V:

${\displaystyle {\dim }_{\mathbb{C}}{V}^{\mathbb{C}}={\dim }_{\mathbb{R}}V.}$

Alternatively, rather than using tensor products, one can use this direct sum as the definition of the complexification:

${\displaystyle V^{\mathbb{C}}:=V\oplus V,}$

where ${\displaystyle V^{\mathbb{C}}}$ is given a linear complex structure by the operator J defined as ${\displaystyle J(v,w):=(-w,v),}$ where J encodes the operation of “multiplication by i”. In matrix form, J is given by:

${\displaystyle J=\left[\begin{array}{cc}0& -I_V\\ I_V& 0\end{array}\right].}$

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, ${\displaystyle V^{\mathbb{C}}}$ can be written as ${\displaystyle V\oplus JV}$ or ${\displaystyle V\oplus iV,}$ 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.

Examples

• The complexification of real coordinate space Rⁿ is the complex coordinate space Cⁿ.
• Likewise, if V consists of the m×n matrices with real entries, V^C would consist of m×n matrices with complex entries.

Dickson doubling

The process of complexification by moving from R to C was abstracted by twentieth-century mathematicians including Leonard Dickson. One starts with using the identity mapping x* = x as a trivial involution on R. Next two copies of R are used to form z = (a , b) with the complex conjugation introduced as the involution z* = (a, −b). Two elements w and z in the doubled set multiply by

${\displaystyle wz=(a,b)\times (c,d)=(ac-d^{\ast }b,da+b{c}^{\ast }).}$

Finally, the doubled set is given a norm N(z) = z* z. When starting from R with the identity involution, the doubled set is C with the norm a² + b². If one doubles C, and uses conjugation (a, b)* = (a*, –b), the construction yields quaternions. Doubling again produces octonions, also called Cayley numbers. It was at this point that Dickson in 1919 contributed to uncovering algebraic structure.

The process can also be initiated with C and the trivial involution z* = z. The norm produced is simply z², unlike the generation of C by doubling R. When this C is doubled it produces bicomplex numbers, and doubling that produces biquaternions, and doubling again results in bioctonions. When the base algebra is associative, the algebra produced by this Cayley–Dickson construction is called a composition algebra since it can be shown that it has the property

${\displaystyle N(pq)=N(p)N(q).}$

Complex conjugation

The complexified vector space V^C has more structure than an ordinary complex vector space. It comes with a canonical complex conjugation map:

${\displaystyle \chi :V^{\mathbb{C}}\to \overline{V^{\mathbb{C}}}}$

defined by

${\displaystyle \chi (v\otimes z)=v\otimes \overline{z}.}$

The map χ may either be regarded as a conjugate-linear map from V^C to itself or as a complex linear isomorphism from V^C to its complex conjugate ${\displaystyle \overline{V^{\mathbb{C}}}}$.

Conversely, given a complex vector space W with a complex conjugation χ, W is isomorphic as a complex vector space to the complexification V^C of the real subspace

${\displaystyle V=\{w\in W:\chi (w)=w\}.}$

In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.

For example, when W = Cⁿ with the standard complex conjugation

${\displaystyle \chi (z_1,\dots ,z_n)=({\overline{z}}_1,\dots ,{\overline{z}}_n)}$

the invariant subspace V is just the real subspace Rⁿ.

Linear transformations

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

${\displaystyle f^{\mathbb{C}}:V^{\mathbb{C}}\to W^{\mathbb{C}}}$

given by

${\displaystyle f^{\mathbb{C}}(v\otimes z)=f(v)\otimes z.}$

The map ${\displaystyle f^{\mathbb{C}}}$ is called the complexification of f. The complexification of linear transformations satisfies the following properties

• ${\displaystyle ({\mathrm{i}\mathrm{d}}_V{)}^{\mathbb{C}}={\mathrm{i}\mathrm{d}}_{V^{\mathbb{C}}}}$
• ${\displaystyle (f\circ g{)}^{\mathbb{C}}=f^{\mathbb{C}}\circ g^{\mathbb{C}}}$
• ${\displaystyle (f+g{)}^{\mathbb{C}}=f^{\mathbb{C}}+g^{\mathbb{C}}}$
• ${\displaystyle (af{)}^{\mathbb{C}}=a{f}^{\mathbb{C}}\mathrm{\forall }a\in \mathbb{R}}$

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 f^C commutes with conjugation and so maps the real subspace of V^C to the real subspace of W^C (via the map f). Moreover, a complex linear map g : V^C → W^C 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 R^m 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 C^m.

Dual spaces and tensor products

The dual of a real vector space V is the space V* of all real linear maps from V to R. The complexification of V* can naturally be thought of as the space of all real linear maps from V to C (denoted Hom_R(V,C)). That is, $${\displaystyle (V^{\ast }{)}^{\mathbb{C}}=V^{\ast }\otimes \mathbb{C}\cong {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathbb{R}}(V,\mathbb{C}).}$$

The isomorphism is given by $${\displaystyle ({\phi }_1\otimes 1+{\phi }_2\otimes i)\leftrightarrow {\phi }_1+i{\phi }_2}$$ where φ₁ and φ₂ are elements of V*. Complex conjugation is then given by the usual operation $${\displaystyle \overline{{\phi }_1+i{\phi }_2}={\phi }_1-i{\phi }_2.}$$

Given a real linear map φ : V → C we may extend by linearity to obtain a complex linear map φ : V^C → C. That is, $${\displaystyle \phi (v\otimes z)=z\phi (v).}$$ This extension gives an isomorphism from Hom_R(V,C) to Hom_C(V^C,C). The latter is just the complex dual space to V^C, so we have a natural isomorphism: $${\displaystyle (V^{\ast }{)}^{\mathbb{C}}\cong (V^{\mathbb{C}}{)}^{\ast }.}$$

More generally, given real vector spaces V and W there is a natural isomorphism $${\displaystyle {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathbb{R}}(V,W{)}^{\mathbb{C}}\cong {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathbb{C}}(V^{\mathbb{C}},W^{\mathbb{C}}).}$$

Complexification also commutes with the operations of taking tensor products, exterior powers and symmetric powers. For example, if V and W are real vector spaces there is a natural isomorphism $${\displaystyle (V{\otimes }_{\mathbb{R}}W{)}^{\mathbb{C}}\cong V^{\mathbb{C}}{\otimes }_{\mathbb{C}}{W}^{\mathbb{C}}.}$$ Note the left-hand tensor product is taken over the reals while the right-hand one is taken over the complexes. The same pattern is true in general. For instance, one has $${\displaystyle ({\mathrm{\Lambda }}_{\mathbb{R}}^{k}V{)}^{\mathbb{C}}\cong {\mathrm{\Lambda }}_{\mathbb{C}}^k(V^{\mathbb{C}}).}$$ In all cases, the isomorphisms are the “obvious” ones.