complexification of vector space
0.1 Complexification of vector space
If $V$ is a real vector space, its complexification ${V}^{\u2102}$ 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:
$(x+iy)+(u+iv)$ | $=(x+u)+i(y+v),$ | $x,y,u,v\in V$ | ||
$(\alpha +i\beta )(x+iy)$ | $=(\alpha x-\beta y)+i(\beta x+\alpha y),$ | $x,y\in V,\alpha ,\beta \in \mathbb{R}.$ |
If ${v}_{1},\mathrm{\dots},{v}_{n}$ is a basis for $V$, then ${v}_{1}+i0,\mathrm{\dots},{v}_{n}+i0$ is a basis for ${V}^{\u2102}$. Naturally, $x+i0\in {V}^{\u2102}$ is often written just as $x$.
So, for example, the complexification of ${\mathbb{R}}^{n}$ is (isomorphic^{} to) ${\u2102}^{n}$.
0.2 Complexification of linear transformation
If $T:V\to W$ is a linear transformation between two real vector spaces $V$ and $W$, its complexification ${T}^{\u2102}:{V}^{\u2102}\to {W}^{\u2102}$ is defined by
${T}^{\u2102}(x+iy)=Tx+iTy.$ |
It may be readily verified that ${T}^{\u2102}$ is complex-linear.
If ${v}_{1},\mathrm{\dots},{v}_{n}$ is a basis for $V$, ${w}_{1},\mathrm{\dots},{w}_{m}$ 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 ${T}^{\u2102}$ with respect to the corresponding bases in ${V}^{\u2102}$ and ${W}^{\u2102}$.
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^{} $T:V\to V$, 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 ${T}^{\u2102}$.
Also, the complexification process generalizes without change for infinite-dimensional spaces.
0.3 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 ${V}^{\u2102}$ by the obvious expansion:
$$\u27e8x+iy,u+iv\u27e9=\u27e8x,u\u27e9+\u27e8y,v\u27e9+i(\u27e8y,u\u27e9-\u27e8x,v\u27e9).$$ |
It follows that ${\parallel x+iy\parallel}^{2}={\parallel x\parallel}^{2}+{\parallel y\parallel}^{2}$.
0.4 Complexification of norm
More generally, for a real normed vector space^{} $V$, the equation
$${\parallel x+iy\parallel}^{2}={\parallel x\parallel}^{2}+{\parallel y\parallel}^{2}$$ |
can serve as a definition of the norm for ${V}^{\u2102}$.
References
- 1 Vladimir I. Arnol’d. Ordinary Differential Equations^{}. Springer-Verlag, 1992.
Title | complexification of vector space |
---|---|
Canonical name | ComplexificationOfVectorSpace |
Date of creation | 2013-03-22 15:24:33 |
Last modified on | 2013-03-22 15:24:33 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 9 |
Author | stevecheng (10074) |
Entry type | Definition |
Classification | msc 15A04 |
Classification | msc 15A03 |
Related topic | ComplexStructure2 |
Defines | complexification |