complexification of vector space

0.1 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 $v_{1},\ldots,v_{n}$ is a basis for $V$ , then $v_{1}+i0,\ldots,v_{n}+i0$ 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) $\mathbb{C}^{n}$ .

0.2 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 $v_{1},\ldots,v_{n}$ is a basis for $V$ , $w_{1},\ldots,w_{m}$ 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.

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 $V^{\mathbb{C}}$ by the obvious expansion:

$\langle x+iy,u+iv\rangle=\langle x,u\rangle+\langle y,v\rangle+i(\langle y,u% \rangle-\langle x,v\rangle)\,.$

It follows that $\lVert x+iy\rVert^{2}=\lVert x\rVert^{2}+\lVert y\rVert^{2}$ .

0.4 Complexification of norm

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

$\lVert x+iy\rVert^{2}=\lVert x\rVert^{2}+\lVert y\rVert^{2}$

can serve as a definition of the norm for $V^{\mathbb{C}}$ .

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