complexification of vector space

0.1 Complexification of vector space

If V is a real vector space, its complexification V is the complex vector space consisting of elements x+iy, where x,yV. Vector addition and scalar multiplication by complex numbersPlanetmathPlanetmath are defined in the obvious way:

(x+iy)+(u+iv) =(x+u)+i(y+v), x,y,u,vV
(α+iβ)(x+iy) =(αx-βy)+i(βx+αy), x,yV,α,β.

If v1,,vn is a basis for V, then v1+i0,,vn+i0 is a basis for V. Naturally, x+i0V is often written just as x.

So, for example, the complexification of n is (isomorphicPlanetmathPlanetmathPlanetmath to) n.

0.2 Complexification of linear transformation

If T:VW is a linear transformation between two real vector spaces V and W, its complexification T:VW is defined by


It may be readily verified that T is complex-linear.

If v1,,vn is a basis for V, w1,,wm is a basis for W, and A is the matrix representationPlanetmathPlanetmath of T with respect to these bases, then A, regarded as a complex matrix, is also the representation of T with respect to the corresponding bases in V and W.

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 coordinatesPlanetmathPlanetmath and use matrix representations when otherwise there is no need to. For example, we might want to make argumentsPlanetmathPlanetmath about the complex eigenvalues and eigenvectors for a transformationPlanetmathPlanetmath T:VV, 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.

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 spaceMathworldPlanetmath, its real inner productMathworldPlanetmath can be extended to a complex inner product for V by the obvious expansion:


It follows that x+iy2=x2+y2.

0.4 Complexification of norm

More generally, for a real normed vector spacePlanetmathPlanetmath V, the equation


can serve as a definition of the norm for V.


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