In mathematics, the complexification of a real vector space V is a vector space V^C 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 V^C over the complex numbers. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... The fundamental concept in linear algebra is that of a vector space or linear space. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

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): In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

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 V^C is only a real vector space. However, we can make V^C into a complex vector space by defining complex multiplication as follows:

Basic properties

By the nature of the tensor product, every vector v in V^C 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 V^C as the direct sum of two copies of V: In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

with the above rule for multiplication by complex numbers.

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

The vector space V may then be regarded as a real subspace of V^C. If V has a basis {e_i} then a corresponding basis for V^C is given by {e_i⊗1}. The complex dimension of V^C is therefore equal to the real dimension of V: This article is about linear subspaces of an abstract vector space. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...

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, V^C would consist of m×n matrices with complex entries.

In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

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: Canonical is an adjective derived from canon. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

defined by

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 . In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, one associates to every vector space V over the complex numbers C its complex conjugate vector space V*, again a vector space over C. The underlying set and the addition of V* are the same as those of V, and the scalar multiplication in V* is defined as...

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

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 In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

given by

The map f^C 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. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In category theory, a branch of mathematics, a functor is a special type of mapping between categories. ... In mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms. ...

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 the exact same matrix, but now thought of as a linear map from Cⁿ to C^m. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

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, In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ...

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 : V^C → C. That is,

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: In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

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 In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ...

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 isomorphism are the “obvious” ones.

See also

• Linear complex structure

In mathematics, a complex structure on a real vector space V is is an real linear transformation J : V → V such that J2 = −idV. Here J2 means J composed with itself and idV is the identity map on V. That is, the effect of applying J twice is the same...

References

• Roman, Steven (2005). Advanced Linear Algebra, (2nd ed.), Graduate Texts in Mathematics 135, New York: Springer. ISBN 0-387-24766-1.