Complexification

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

definition

There are two different ways of defining the complexification of a real vector space. The two possibilities that are now presented are equivalent.

Using the direct sum

Let be a vector space over the field of real numbers . The complexification of is the direct sum ${\displaystyle V}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle V}$

${\displaystyle V_{\mathbb {C} }=V\oplus V=V\times V.}$

In the new space the addition becomes component-wise

${\displaystyle (x,\,y)+(x',\,y')=(x+x',\,y+y')}$

and the scalar multiplication with by ${\displaystyle \alpha +\beta i\in \mathbb {C} }$

${\displaystyle (\alpha +\beta i)(x,\,y)=\left(\alpha x-\beta y,\,\beta x+\alpha y\right)}$

Are defined.

This turns into a vector space over the body of complex numbers . ${\displaystyle V_{\mathbb {C} }}$ ${\displaystyle \mathbb {C} }$

In analogy to the notation of complex numbers, one also writes for the pair . ${\displaystyle (x,y)\in V_{\mathbb {C} }}$ ${\displaystyle x+yi}$

By means of the tensor product

The complexification can also be defined by the tensor product :

${\displaystyle V_{\mathbb {C} }=V\otimes _{\mathbb {R} }\mathbb {C} }$.

Then the scalar multiplication is given by by, i.e. i.e., for with and applies ${\displaystyle a\in \mathbb {C} }$ ${\displaystyle \operatorname {id} \otimes a}$ ${\displaystyle x\otimes b\in V_{\mathbb {C} }}$ ${\displaystyle x\in V}$ ${\displaystyle b\in \mathbb {C} }$

${\displaystyle a(x\otimes b)=x\otimes (ab)}$.

Examples

• The complexification of Euclidean space results in the unitary space . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$
• The Complexification of the vector space of - matrices with real entries gives the vector space of matrices with complex entries. Complexification thus abstracts the simple fact that real numbers can in particular also be understood as complex numbers. ${\displaystyle \mathbb {R} ^{m\times n}}$ ${\displaystyle m\times n}$ ${\displaystyle \mathbb {C} ^{m\times n}}$

properties

• The real vector space can be understood as a real sub-vector space of by means of the embedding . It is in if and only if applies. ${\displaystyle V}$ ${\displaystyle x\mapsto x+0i}$ ${\displaystyle V_{\mathbb {C} }}$ ${\displaystyle x+yi\in V_{\mathbb {C} }}$ ${\displaystyle V}$ ${\displaystyle y=0}$
• In a natural way involution defined that the complex conjugation corresponds. A is in if and only applies. ${\displaystyle V_{\mathbb {C} }}$ ${\displaystyle {\overline {x+yi}}=x-yi}$ ${\displaystyle z\in V_{\mathbb {C} }}$ ${\displaystyle V}$ ${\displaystyle {\overline {z}}=z}$
• Is a basis of , then is a basis of- vector space . In particular, the real vector space and the complex vector space have the same dimension . ${\displaystyle (x_{j})}$ ${\displaystyle V}$ ${\displaystyle (x_{j}+i0)}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle V_{\mathbb {C} }}$ ${\displaystyle V}$ ${\displaystyle V_{\mathbb {C} }}$

Complexification of linear maps

definition

Each -linear mapping provides a -linear mapping defined by ${\displaystyle \mathbb {R} }$ ${\displaystyle f\colon V\rightarrow W}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f_{\mathbb {C} }\colon V_{\mathbb {C} }\rightarrow W_{\mathbb {C} }}$

${\displaystyle f_{\mathbb {C} }(x+yi)=f(x)+f(y)i.}$

properties

The following applies to the complexified mapping : ${\displaystyle f_{\mathbb {C} }\colon V_{\mathbb {C} }\rightarrow W_{\mathbb {C} }}$

• ${\displaystyle f_{\mathbb {C} }({\overline {z}})={\overline {f_{\mathbb {C} }(z)}}}$ for all ${\displaystyle z\in V_{\mathbb {C} }}$
• ${\displaystyle (\operatorname {Id} _{V})_{\mathbb {C} }=\operatorname {Id} _{(V_{\mathbb {C} })}}$
• ${\displaystyle (f\circ g)_{\mathbb {C} }=f_{\mathbb {C} }\circ g_{\mathbb {C} }}$
• The representative matrix of relative to the base is equal to the representative matrix of relative to the base . ${\displaystyle f}$ ${\displaystyle (x_{j})}$ ${\displaystyle f_{\mathbb {C} }}$ ${\displaystyle (x_{j}+0i)}$

If the linear mapping to be considered is an endomorphism , then the following also applies: ${\displaystyle f_{\mathbb {C} }\colon V_{\mathbb {C} }\rightarrow V_{\mathbb {C} }}$

• ${\displaystyle f}$and have the same characteristic polynomial . ${\displaystyle f_{\mathbb {C} }}$
• ${\displaystyle f_{\mathbb {C} }}$ has all eigenvalues ​​of f.

Complex matrices are often easier to describe than the real original. For example, every complex matrix can be trigonalized , and the normal matrices mentioned above can even be diagonalized .

Complexification of bilinear forms and scalar products

definition

For a bilinear form there is a sesquilinear form given by ${\displaystyle \Phi \colon V\times V\rightarrow \mathbb {R} }$ ${\displaystyle \Phi _{\mathbb {C} }\colon V_{\mathbb {C} }\times V_{\mathbb {C} }\rightarrow \mathbb {C} }$

${\displaystyle \Phi _{\mathbb {C} }(x+yi,x'+y'i)=\Phi (x,x')+\Phi (y,y')+i(\Phi (y,x')-\Phi (x,y')).}$

It applies , the restriction of on is again . ${\displaystyle \Phi _{\mathbb {C} |V\times V}=\Phi }$ ${\displaystyle \Phi _{\mathbb {C} }}$ ${\displaystyle V\times V}$ ${\displaystyle \Phi }$

properties

• The form is a real scalar product if and only if is a complex scalar product. Since the complex scalar product is easier to describe than the real one, it is complexified in order to then continue working in complex space. ${\displaystyle \Phi }$ ${\displaystyle \Phi _{\mathbb {C} }}$

• If V is Euclidean with scalar product and the associated unitary vector space with scalar product then applies . That is, the operation of complexifying the adjunction can be reversed. It follows from this that the complexification acquires certain properties of a linear mapping. The mapping has one of the following properties if and only if it has: ${\displaystyle \Phi }$ ${\displaystyle V_{\mathbb {C} }}$ ${\displaystyle \Phi _{\mathbb {C} }}$ ${\displaystyle (f^{*})_{\mathbb {C} }=(f_{\mathbb {C} })^{*}}$ ${\displaystyle f\colon V\rightarrow V}$ ${\displaystyle f_{\mathbb {C} }\colon V_{\mathbb {C} }\rightarrow V_{\mathbb {C} }}$
  • normal ${\displaystyle (ff^{*}=f^{*}f)}$
  • self adjoint ${\displaystyle (f=f^{*})}$
  • asymmetrically ${\displaystyle (f=-f^{*})}$
  • Isometry ${\displaystyle (ff^{*}=\operatorname {Id} )}$

Complexification of a Lie algebra

definition

Let it be a Lie algebra over the body . The complexification of the Lie algebra is the Lie algebra , which is analogous to the complexified vector space by ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }}$

${\displaystyle {\mathfrak {g}}_{\mathbb {C} }:={\mathfrak {g}}\otimes _{\mathbb {R} }\mathbb {C} }$

is defined.

The complexification of a Lie algebra can also be understood as an extension of the underlying field of the Lie algebra to the field . An element of Lie algebra can be understood as a pair with . The operations on are then defined by ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }}$ ${\displaystyle (u,v)}$ ${\displaystyle u,v\in {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }}$

${\displaystyle {\begin{aligned}&(u_{1},v_{1})+(u_{2},v_{2}):=(u_{1}+u_{2},v_{1}+v_{2})\\&(\alpha +i\beta )(u_{1},v_{1}):=(\alpha u_{1}-\beta v_{1},\alpha v_{1}+\beta u_{1})\quad {\text{und}}\\&\left[(u_{1},v_{1}),(u_{2},v_{2})\right]:=(\left[u_{1},u_{2}\right]-\left[v_{1},v_{2}\right],\left[v_{1},u_{2}\right]+\left[u_{1},v_{2}\right]),\end{aligned}}}$

where and applies. In addition, the addition and the Lie bracket are in Lie algebra. ${\displaystyle \alpha ,\beta \in \mathbb {R} }$ ${\displaystyle u_{1},u_{2},v_{1},v_{2}\in {\mathfrak {g}}}$ ${\displaystyle +}$ ${\displaystyle \left[.,.\right]}$

Examples

• The complexification of is . ${\displaystyle \mathrm {sl} (n,\mathbb {R} )=\left\{A\in \mathrm {Mat} (n,\mathbb {R} ):\mathrm {Tr} (A)=0\right\}}$ ${\displaystyle \mathrm {sl} (n,\mathbb {C} )=\left\{A\in \mathrm {Mat} (n,\mathbb {C} ):\mathrm {Tr} (A)=0\right\}}$
• The Cartan decomposition has for the shape ${\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {g}}=\mathrm {sl} (n,\mathbb {C} )}$

${\displaystyle {\mathfrak {k}}=\left\{A\in \mathrm {sl} (n,\mathbb {C} ):A{\text{ schiefhermitesch}}\right\},{\mathfrak {p}}=\left\{A\in \mathrm {sl} (n,\mathbb {C} ):A{\text{ hermitesch}}\right\}}$,

which follows in this special case and with it. ${\displaystyle {\mathfrak {p}}=i{\mathfrak {k}}}$ ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }={\mathfrak {g}}\oplus {\mathfrak {g}}}$

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 . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}_{\mathbb {C} }}$

In general, if not simply connected, a complex Lie group is called the complexification of , if there is a continuous homomorphism with the following universal property: for every continuous homomorphism in a complex Lie group there is a unique complex-analytic homomorphism with . Complexification does not always have to exist, but it is clear when it exists. ${\displaystyle G}$ ${\displaystyle G_{\mathbb {C} }}$ ${\displaystyle G}$ ${\displaystyle \phi :G\rightarrow G_{\mathbb {C} }}$ ${\displaystyle f:G\rightarrow H}$ ${\displaystyle H}$ ${\displaystyle F:G_{\mathbb {C} }\rightarrow H}$ ${\displaystyle f=F\phi }$

Examples: The complexification of is , the complexification of is , the complexification of is . ${\displaystyle SL(n,\mathbb {R} )}$ ${\displaystyle SL(n,\mathbb {C} )}$ ${\displaystyle SU(n)}$ ${\displaystyle SL(n,\mathbb {C} )}$ ${\displaystyle SL(n,\mathbb {C} )}$ ${\displaystyle SL(n,\mathbb {C} )\times SL(n,\mathbb {C} )}$

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 to the category of vector spaces over the complex numbers. The morphisms of the categories are in each case the -linear mappings, whereby the following applies for the real and for the complex vector spaces. The functor adjoint to this functor on the right is the forget functor from the category of complex vector spaces to the category of real vector spaces, which "forgets" the complex structure of the spaces. ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$

literature

• Theodor Bröcker: Linear Algebra and Analytical Geometry , Birkhäuser Verlag, 2004, ISBN 3-7643-7144-7 .
• Gerd Fischer: Lineare Algebra , Vieweg-Verlag, ISBN 3-528-97217-3 .
• VL Popov: Complexification of a Lie algebra . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).