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complexification of vector space (Definition)

Complexification of vector space

If $V$ is a real vector space, its complexification $\Vc$ 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 $\Vc$ . Naturally, $x + i0 \in \Vc$ is often written just as $x$ .

So, for example, the complexification of $\real^n$ is (isomorphic to) $\complex^n$ .

Complexification of linear transformation

If $T\colon V \to W$ is a linear transformation between two real vector spaces $V$ and $W$ , its complexification $\Tc\colon \Vc \to \Wc$ is defined by

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

It may be readily verified that $\Tc$ 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 $\Tc$ with respect to the corresponding bases in $\Vc$ and $\Wc$ .

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 $\Tc$ .

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 $\Vc$ 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 $\norm{x+iy}^2 = \norm{x}^2 + \norm{y}^2$ .

Complexification of norm

More generally, for a real normed vector space $V$ , the equation $$ \norm{x+iy}^2 = \norm{x}^2 + \norm{y}^2 $$ can serve as a definition of the norm for $\Vc$ .

Bibliography

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




"complexification of vector space" is owned by stevecheng.
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See Also: linear complex structure

Also defines:  complexification
Keywords:  complex matrix
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Cross-references: norm, equation, normed vector space, inner product, inner product space, infinite-dimensional, transformation, eigenvectors, eigenvalues, coordinates, matrix, matrix representation, linear transformation, isomorphic, basis, obvious, complex numbers, multiplication, scalar, vector addition, elements, complex, vector space, real
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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 3997 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)

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