Complexification: Difference between revisions
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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 |
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 |
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:<math>(\Lambda_{\mathbb R}^k V)^{\mathbb C} \cong \Lambda_{\mathbb C}^k (V^{\mathbb C}).</math> |
:<math>(\Lambda_{\mathbb R}^k V)^{\mathbb C} \cong \Lambda_{\mathbb C}^k (V^{\mathbb C}).</math> |
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In all cases, the |
In all cases, the isomorphisms are the “obvious” ones. |
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==See also== |
==See also== |
Revision as of 08:37, 23 January 2010
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex numbers.
Formal definition
Let V be a real vector space. The complexification of V is defined by taking the tensor product of V with the complex numbers (thought of as a two-dimensional vector space over the reals):
The subscript R on the tensor product indicates that the tensor product is taken over the real numbers (since V is a real vector space this is the only sensible option anyway, so the subscript can safely be omitted). As it stands VC is only a real vector space. However, we can make VC into a complex vector space by defining complex multiplication as follows:
Basic properties
By the nature of the tensor product, every vector v in VC can be written uniquely in the form
where v1 and v2 are vectors in V. It is a common practice to drop the tensor product symbol and just write
Multiplication by the complex number a + ib is then given by the usual rule
We can then regard VC as the direct sum of two copies of V:
with the above rule for multiplication by complex numbers.
There is a natural embedding of V into VC given by
The vector space V may then be regarded as a real subspace of VC. If V has a basis {ei} then a corresponding basis for VC is given by {ei⊗1}. The complex dimension of VC is therefore equal to the real dimension of V:
Examples
- The complexification of real coordinate space Rn is complex coordinate space Cn.
- Likewise, if V consists of the m×n matrices with real entries, VC would consist of m×n matrices with complex entries.
Complex conjugation
The complexified vector space VC has more structure than an ordinary complex vector space. It comes with a canonical complex conjugation map:
defined by
The map χ may either be regarded as a conjugate-linear map from VC to itself or as a complex linear isomorphism from VC to its complex conjugate .
Conversely, given a complex vector space W with a complex conjugation χ, W is isomorphic as a complex vector space to the complexification VC of the real subspace
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
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
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 the exact same matrix, but now thought of as a linear map from Cn to Cm.
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 HomR(V,C)). That is,
The isomorphism is given by
where φ1 and φ2 are elements of V*. Complex conjugation is then given by the usual operation
Given a real linear map φ : V → C we may extend by linearity to obtain a complex linear map φ : VC → C. That is,
This extension gives an isomorphism from HomR(V,C)) to HomC(VC,C). The latter is just the complex dual space to VC, so we have a natural isomorphism:
More generally, given real vector spaces V and W there is a natural isomorphism
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
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
In all cases, the isomorphisms are the “obvious” ones.
See also
References
- Roman, Steven (2005). Advanced Linear Algebra. Graduate Texts in Mathematics 135 ((2nd ed.) ed.). New York: Springer. ISBN 0-387-24766-1.