complexification of vector space

0.1 Complexification of vector space

If $V$ is a real vector space,its complexification $V^{ℂ}$is the complex vector space consisting of elements $x+iy$, where $x,y\in V$. Vector addition and scalar multiplication by complex numbersare 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 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 $V^{ℂ}$. Naturally, $x+i0\in V^{ℂ}$ is often written just as $x$.

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

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 $T^{ℂ}:V^{ℂ}\to W^{ℂ}$is defined by

$T^{ℂ}(x+iy)=Tx+iTy.$

It may be readily verified that $T^{ℂ}$ 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 $T^{ℂ}$ with respect to the corresponding basesin $V^{ℂ}$ and $W^{ℂ}$.

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 thatwe are not required to fix a choice of coordinates and use matrix representationswhen otherwise there is no need to.For example, we might want to make arguments about the complex eigenvaluesand eigenvectors for a transformation $T:V\to V$, while,of course, non-real eigenvalues and eigenvectors, by definition, cannot existfor a transformation between real vector spaces.What we really mean are the eigenvalues and eigenvectors of $T^{ℂ}$.

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

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 $V^{ℂ}$ bythe obvious expansion:

$$⟨x+iy,u+iv⟩=⟨x,u⟩+⟨y,v⟩+i(⟨y,u⟩-⟨x,v⟩).$$

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 $V^{ℂ}$.