PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: High Entry average rating: No information on entry rating
complexification of vector space (Definition)

Complexification of vector space

If $ V$ is a real vector space, its complexification $ V^\mathbb{C}$ is the complex vector space consisting of elements $ x + iy$, where $ x, y \in V$. Vector addition and scalar multiplication by complex numbers are defined in the obvious way:

$\displaystyle (x + iy) + (u+iv)$ $\displaystyle = (x+u) + i(y+v)\,,$ $\displaystyle x, y, u, v \in V$    
$\displaystyle (\alpha + i\beta) (x+iy)$ $\displaystyle = (\alpha x - \beta y) + i(\beta x + \alpha y),$ $\displaystyle x, y \in V, \alpha, \beta \in \mathbb{R}\,.$    

If $ v_1, \dotsc, v_n$ is a basis for $ V$, then $ v_1 + i0, \dotsc, v_n + i0$ is a basis for $ V^\mathbb{C}$. Naturally, $ x + i0 \in V^\mathbb{C}$ is often written just as $ x$.

So, for example, the complexification of $ \mathbb{R}^n$ is (isomorphic to) $ \mathbb{C}^n$.

Complexification of linear transformation

If $ T\colon V \to W$ is a linear transformation between two real vector spaces $ V$ and $ W$, its complexification $ T^\mathbb{C}\colon V^\mathbb{C}\to W^\mathbb{C}$ is defined by

$\displaystyle T^\mathbb{C}(x+iy) = Tx + iTy\,.$    

It may be readily verified that $ T^\mathbb{C}$ is complex-linear.

If $ v_1, \dotsc, v_n$ is a basis for $ V$, $ w_1, \dotsc, w_m$ is a basis for $ W$, and $ A$ is the matrix representation of $ T$ with respect to these bases, then $ A$, regarded as a complex matrix, is also the representation of $ T^\mathbb{C}$ with respect to the corresponding bases in $ V^\mathbb{C}$ and $ W^\mathbb{C}$.

So, the complexification process is a formal, coordinate-free way of saying: take the matrix $ A$ of $ T$, with its real entries, but operate on it as a complex matrix. The advantage of making this abstracted definition is that we are not required to fix a choice of coordinates and use matrix representations when otherwise there is no need to. For example, we might want to make arguments about the complex eigenvalues and eigenvectors for a transformation $ T\colon V \to V$, while, of course, non-real eigenvalues and eigenvectors, by definition, cannot exist for a transformation between real vector spaces. What we really mean are the eigenvalues and eigenvectors of $ T^\mathbb{C}$.

Also, the complexification process generalizes without change for infinite-dimensional spaces.

Complexification of inner product

Finally, if $ V$ is also a real inner product space, its real inner product can be extended to a complex inner product for $ V^\mathbb{C}$ by the obvious expansion:

$\displaystyle \langle x+iy, u+iv \rangle = \langle x, u \rangle + \langle y, v \rangle + i(\langle y, u \rangle - \langle x, v \rangle)\,. $
It follows that $ \lVert x+iy\rVert ^2 = \lVert x\rVert ^2 + \lVert y\rVert ^2$.

Complexification of norm

More generally, for a real normed vector space $ V$, the equation

$\displaystyle \lVert x+iy\rVert ^2 = \lVert x\rVert ^2 + \lVert y\rVert ^2 $
can serve as a definition of the norm for $ V^\mathbb{C}$.

Bibliography

1
Vladimir I. Arnol'd. Ordinary Differential Equations. Springer-Verlag, 1992.



"complexification of vector space" is owned by stevecheng.
(view preamble)

View style:

See Also: linear complex structure

Also defines:  complexification
Keywords:  complex matrix
Log in to rate this entry.
(view current ratings)

Cross-references: norm, equation, normed vector space, inner product, inner product space, infinite-dimensional, transformation, eigenvalues, coordinates, matrix, matrix representation, linear transformation, isomorphic, basis, obvious, complex numbers, multiplication, scalar, vector addition, complex, vector space, real
There are 5 references to this entry.

This is version 6 of complexification of vector space, born on 2005-07-21, modified 2007-06-30.
Object id is 7249, canonical name is ComplexificationOfVectorSpace.
Accessed 2665 times total.

Classification:
AMS MSC15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
 15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)