Complexification

In linear algebra, a complexification is an operation that maps a real vector space a complex vector space which has very similar properties.

• 4.1 Definition
• 4.2 Properties

• 5.1 Definition
• 5.2 Properties

• 6.1 Definition
• 6.2 Examples

Definition

There are two different ways the complexification of a real vector space defined. The two possibilities are discussed next, are equivalent.

By means of the direct sum

Be a vector space over the field of real numbers. The complexification of the direct sum

In the new room addition is componentwise

And the scalar multiplication with by

Defined.

This makes it a vector space over the field of complex numbers.

In analogy to the notation of complex numbers to write for the couple as well.

By means of the tensor product

It is the complexification also define the tensor product:

Then, the scalar multiplication is added with by, i.e., with and for valid

Examples

• The complexification of the Euclidean space yields the unitary space.
• The complexification of the vector space of matrices with real entries yields the vector space of matrices with complex entries. Thus the complexification abstracts the simple fact that one particular may be regarded as complex numbers real numbers.

Properties

• The real vector space can be set via the embedding as a real subspace of understand. Here then is exactly in if applies.
• An involution is defined in a natural way, which corresponds to the complex conjugate. A iff is if applies.
• Is a base of such a base of the vector space. In particular, the real vector space, and the complex vector space have the same dimension.

Complexification linear maps

Definition

Each linear map provides a linear map defined by

Properties

For the complexified figure:

• For all
• The matrix representing of respect to the base is equal to the representative matrix of relative to the base.

Is to be considered linear map is an endomorphism, then also applies:

• And have the same characteristic polynomial.
• Has all eigenvalues ​​of f

Complexified matrices are often easier to describe than the real original. For example, any complex matrix trigonalisierbar, wherein the above -mentioned normal matrices can even be diagonalized.

Complexification of bilinear forms and scalar products

Definition

There is a sesquilinear form given to a bilinear form by

It is true, the restriction of to is thus again.

Properties

• The form is then exactly a real scalar, if a complex scalar product is. Since the complex scalar product is easier to describe than the real, you komplexifiziert it, and then continue working in the complex space.

• If V is euclidean with scalar product and the corresponding unitary vector space with scalar product shall apply. That is, the operation of the complexification of the adjunction can be interchanged. It follows that the complexification receives certain properties of a linear map. The figure has exactly then one of the following, even if it has: normal
• Self-adjoint
• Skew-symmetric
• Isometry

Complexification of a Lie algebra

Definition

It is a Lie algebra to the body. The complexification of the Lie algebra is the Lie algebra, which by analogy to the complexified vector space

Is defined.

The complexification of a Lie algebra can be considered an extension of the underlying body of the Lie algebra of the body. One element of the Lie algebra can be understood as a pair with. The operations are then defined by

Where and are valid. Moreover, the addition and the Lie bracket in the Lie algebra.

Examples

• The complexification of is.
• The Cartan decomposition has the form

Resulting in this particular case, and thus follows.

Complexification of a Lie group

The complexification of a simply connected Lie group with Lie algebra is, by definition, the (uniquely determined ) simply connected Lie group with Lie algebra.

General, if it is not simply connected, ie, a complex Lie group the complexification of, if there is a continuous homomorphism with the following universal property: for every continuous homomorphism into a complex Lie group, there is a unique complex analytic homomorphism with. The complexification does not always exist, but it is unique if it exists.

Examples: The complexification of is, is the complexification of.

Category theory

In the language of category theory, the complexification of vector spaces is a functor from the category of vector spaces over the real numbers in the category of vector spaces over the complex numbers. The morphisms of the categories are respectively the - linear maps, where for the real and the complex vector spaces. The right adjoint to this functor is functor of forget - functor from the category of complex vector spaces to the category of real vector spaces, which "forgets" the complex structure of the rooms.