Complexification
Encyclopedia
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.
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:
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
VectR → VectC, 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
VectC → VectR from forgetting the complex structure.
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}(over the field R) then a corresponding basis for VC is given by {ei⊗1} over the field C. The complex dimension of VC is therefore equal to the real dimension of V:
Alternatively, rather than using tensor products, one can use this direct sum as the definition of the complexification:
where
is given a linear complex structure
by the operator J defined as
where J encodes the data of "multiplication by i". In matrix form, J is given by:
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,
can be written as 
or 
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.
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.
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.
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 product
s, 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the complexification of a real vector space V is a vector space VC over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
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 productTensor product
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, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
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:
More generally, complexification is an example of extension of scalars
Extension of scalars
In abstract algebra, extension of scalars is a means of producing a module over a ring S from a module over another ring R, given a homomorphism f : R \to S between them...
– here extending scalars from the real numbers to the complex numbers – which can be done for any field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
, or indeed for any morphism of rings.
Formally, complexification is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
VectR → VectC, 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
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
VectC → VectR from forgetting the complex structure.
Basic properties
By the nature of the tensor product, every vector v in VC can be written uniquely in the formwhere 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
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of VC. If V has a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
{ei}(over the field R) then a corresponding basis for VC is given by {ei⊗1} over the field C. The complex dimension of VC is therefore equal to the real dimension of V:
Alternatively, rather than using tensor products, one can use this direct sum as the definition of the complexification:
where
is given a linear complex structureLinear complex structure
In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.Complex structures have...
by the operator J defined as
where J encodes the data of "multiplication by i". In matrix form, J is given by: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,
can be written as or 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 complex coordinate space Cn.
- Likewise, if V consists of the m×n matricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
with real entries, VC would consist of m×n matrices with complex entries. - The complexification of quaternionQuaternionIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s is the biquaternions. - The complexification of the split-complex numberSplit-complex numberIn abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...
s is the tessarineTessarineIn mathematics, a tessarine is a hypercomplex number of the formt = w + x i + y j + z k, \quad w, x, y, z \in Rwhere i j = j i = k, \quad i^2 = -1, \quad j^2 = +1 .The tessarines are best known for their subalgebra of real tessarines t = w + y j \ ,...
s.
Complex conjugation
The complexified vector space VC has more structure than an ordinary complex vector space. It comes with a canonicalCanonical
Canonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
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
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
from VC to its complex conjugate
Complex conjugate vector space
In mathematics, the complex conjugate of a complex vector space V\, is the complex vector space \overline V consisting of all formal complex conjugates of elements of V\,...
.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 transformationLinear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
one says that complexification defines an (additive) functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from the category of real vector spaces
Category of vector spaces
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...
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
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. 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 dualDual space
In mathematics, any vector space, V, has a corresponding dual vector space 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...
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 product
Tensor product
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, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
s, 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
- Extension of scalarsExtension of scalarsIn abstract algebra, extension of scalars is a means of producing a module over a ring S from a module over another ring R, given a homomorphism f : R \to S between them...
– general process - Linear complex structureLinear complex structureIn mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.Complex structures have...
- Quaternionification – analogous process for quaternions