Complexification

Template:Short description Template:For In mathematics, the complexification of a vector space Template:Math over the field of real numbers (a "real vector space") yields a vector space Template:Math over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for Template:Math (a space over the real numbers) may also serve as a basis for Template:Math over the complex numbers.

Formal definition

Let <math>V</math> be a real vector space. The Template:Em of Template:Math is defined by taking the tensor product of <math>V</math> with the complex numbers (thought of as a 2-dimensional vector space over the reals):

<math>V^{\Complex} = V\otimes_{\R} \Complex\,.</math>

The subscript, <math>\R</math>, on the tensor product indicates that the tensor product is taken over the real numbers (since <math>V</math> is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands, <math>V^{\Complex}</math> is only a real vector space. However, we can make <math>V^{\Complex}</math> into a complex vector space by defining complex multiplication as follows:

<math>\alpha(v \otimes \beta) = v\otimes(\alpha\beta)\qquad\mbox{ for all } v\in V \mbox{ and }\alpha,\beta \in \Complex.</math>

More generally, complexification is an example of extension of scalars – here extending scalars from the real numbers to the complex numbers – which can be done for any field extension, or indeed for any morphism of rings.

Formally, complexification is a functor Template:Math, from the category of real vector spaces to the category of complex vector spaces. This is the adjoint functor – specifically the left adjoint – to the forgetful functor Template:Math forgetting the complex structure.

This forgetting of the complex structure of a complex vector space <math>V</math> is called Template:Em (or sometimes "Template:Em"). The decomplexification of a complex vector space <math>V</math> with basis <math>e_{\mu}</math> removes the possibility of complex multiplication of scalars, thus yielding a real vector space <math>W_{\R}</math> of twice the dimension with a basis <math>\{e_{\mu}, ie_{\mu}\}.</math><ref>Template:Cite book</ref>

Basic properties

By the nature of the tensor product, every vector Template:Math in Template:Math can be written uniquely in the form

<math>v = v_1\otimes 1 + v_2\otimes i</math>

where Template:Math and Template:Math are vectors in Template:Math. It is a common practice to drop the tensor product symbol and just write

<math>v = v_1 + iv_2.\,</math>

Multiplication by the complex number Template:Math is then given by the usual rule

<math>(a+ib)(v_1 + iv_2) = (av_1 - bv_2) + i(bv_1 + av_2).\,</math>

We can then regard Template:Math as the direct sum of two copies of Template:Math:

<math>V^{\Complex} \cong V \oplus i V</math>

with the above rule for multiplication by complex numbers.

There is a natural embedding of Template:Math into Template:Math given by

<math>v\mapsto v\otimes 1.</math>

The vector space Template:Math may then be regarded as a real subspace of Template:Math. If Template:Math has a basis Template:Math (over the field Template:Math) then a corresponding basis for Template:Math is given by Template:Math over the field Template:Math. The complex dimension of Template:Math is therefore equal to the real dimension of Template:Math:

<math>\dim_{\Complex} V^{\Complex} = \dim_{\R} V.</math>

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

<math>V^{\Complex} := V \oplus V,</math>

where <math>V^{\Complex}</math> is given a linear complex structure by the operator Template:Math defined as <math>J(v,w) := (-w,v),</math> where Template:Math encodes the operation of “multiplication by Template:Mvar”. In matrix form, Template:Math is given by:

<math>J = \begin{bmatrix}0 & -I_V \\ I_V & 0\end{bmatrix}.</math>

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, <math>V^{\Complex}</math> can be written as <math>V \oplus JV</math> or <math>V \oplus i V,</math> identifying Template:Math 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 Template:Math is the complex coordinate space Template:Math.
• Likewise, if Template:Math consists of the Template:Math matrices with real entries, Template:Math would consist of Template:Math matrices with complex entries.

Dickson doubling

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The process of complexification by moving from Template:Math to Template:Math was abstracted by twentieth-century mathematicians including Leonard Dickson. One starts with using the identity mapping Template:Math as a trivial involution on Template:Math. Next two copies of R are used to form Template:Math with the complex conjugation introduced as the involution Template:Math. Two elements Template:Mvar and Template:Mvar in the doubled set multiply by

<math>w z = (a,b) \times (c,d) = (ac\ - \ d^*b,\ da \ + \ b c^*).</math>

Finally, the doubled set is given a norm Template:Math. When starting from Template:Math with the identity involution, the doubled set is Template:Math with the norm Template:Math. If one doubles Template:Math, 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 Template:Math and the trivial involution Template:Math. The norm produced is simply Template:Math, unlike the generation of Template:Math by doubling Template:Math. When this Template:Math 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

<math>N(p\,q) = N(p)\,N(q)\,.</math>

Complex conjugation

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

<math>\chi : V^{\Complex} \to \overline{V^{\Complex}}</math>

defined by

<math>\chi(v\otimes z) = v\otimes \bar z.</math>

The map Template:Mvar may either be regarded as a conjugate-linear map from Template:Math to itself or as a complex linear isomorphism from Template:Math to its complex conjugate <math>\overline {V^{\Complex}}</math>.

Conversely, given a complex vector space Template:Math with a complex conjugation Template:Mvar, Template:Math is isomorphic as a complex vector space to the complexification Template:Math of the real subspace

<math>V = \{ w \in W : \chi(w) = w \}.</math>

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

For example, when Template:Math with the standard complex conjugation

<math>\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n)</math>

the invariant subspace Template:Math is just the real subspace Template:Math.

Linear transformations

Given a real linear transformation Template:Math between two real vector spaces there is a natural complex linear transformation

<math>f^{\Complex} : V^{\Complex} \to W^{\Complex}</math>

given by

<math>f^{\Complex}(v\otimes z) = f(v)\otimes z.</math>

The map <math>f^{\Complex}</math> is called the complexification of f. The complexification of linear transformations satisfies the following properties

• <math>(\mathrm{id}_V)^{\Complex} = \mathrm{id}_{V^{\Complex}}</math>
• <math>(f \circ g)^{\Complex} = f^{\Complex} \circ g^{\Complex}</math>
• <math>(f+g)^{\Complex} = f^{\Complex} + g^{\Complex}</math>
• <math>(a f)^{\Complex} = a f^{\Complex} \quad \forall a \in \R</math>

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 Template:Math commutes with conjugation and so maps the real subspace of VTemplate:I sup to the real subspace of Template:Math (via the map Template:Math). Moreover, a complex linear map Template:Math is the complexification of a real linear map if and only if it commutes with conjugation.

As an example consider a linear transformation from Template:Math to Template:Math thought of as an Template:Math matrix. The complexification of that transformation is exactly the same matrix, but now thought of as a linear map from Template:Math to Template:Math.

Dual spaces and tensor products

The dual of a real vector space Template:Math is the space Template:Math of all real linear maps from Template:Math to Template:Math. The complexification of Template:Math can naturally be thought of as the space of all real linear maps from Template:Math to Template:Math (denoted Template:Math). That is, <math display=block>(V^*)^{\Complex} = V^*\otimes \Complex \cong \mathrm{Hom}_{\Reals}(V,\Complex).</math>

The isomorphism is given by <math display=block>(\varphi_1\otimes 1 + \varphi_2\otimes i) \leftrightarrow \varphi_1 + i \varphi_2</math> where Template:Math and Template:Math are elements of Template:Math. Complex conjugation is then given by the usual operation <math display=block>\overline{\varphi_1 + i\varphi_2} = \varphi_1 - i \varphi_2.</math>

Given a real linear map Template:Math we may extend by linearity to obtain a complex linear map Template:Math. That is, <math display=block>\varphi(v\otimes z) = z\varphi(v).</math> This extension gives an isomorphism from Template:Math to Template:Math. The latter is just the complex dual space to Template:Math, so we have a natural isomorphism: <math display=block>(V^*)^{\Complex} \cong (V^{\Complex})^*.</math>

More generally, given real vector spaces Template:Math and Template:Math there is a natural isomorphism <math display=block>\mathrm{Hom}_{\Reals}(V,W)^{\Complex} \cong \mathrm{Hom}_{\Complex}(V^{\Complex},W^{\Complex}).</math>

Complexification also commutes with the operations of taking tensor products, exterior powers and symmetric powers. For example, if Template:Math and Template:Math are real vector spaces there is a natural isomorphism <math display=block>(V \otimes_{\Reals} W)^{\Complex} \cong V^{\Complex} \otimes_{\Complex} W^{\Complex}\,.</math> 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 <math display=block>(\Lambda_{\Reals}^k V)^{\Complex} \cong \Lambda_{\Complex}^k (V^{\Complex}).</math> In all cases, the isomorphisms are the “obvious” ones.

See also

• Extension of scalars – general process
• Linear complex structure
• Baker–Campbell–Hausdorff formula

References

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