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complexification of vector space

(Definition)

Complexification of vector space

If is a real vector space, its complexification $V^\mathbb{C}$ is the complex vector space consisting of elements , 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 , 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 .

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 and , 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 , $w_1, \dotsc, w_m$ is a basis for , and is the matrix representation of with respect to these bases, then , 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 of , 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 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 , 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.

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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, 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 MSC:	15A03 (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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