complexification of vector space
0.1 Complexification of vector space
If is a real vector space, its complexification is the complex vector space consisting of elements , where . Vector addition and scalar multiplication by complex numbers are defined in the obvious way:
If is a basis for , then is a basis for . Naturally, is often written just as .
So, for example, the complexification of is (isomorphic to) .
0.2 Complexification of linear transformation
If is a linear transformation between two real vector spaces and , its complexification is defined by
It may be readily verified that is complex-linear.
If is a basis for , 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 with respect to the corresponding bases in and .
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 , 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 .
Also, the complexification process generalizes without change for infinite-dimensional spaces.
0.3 Complexification of inner product
0.4 Complexification of norm
- 1 Vladimir I. Arnol’d. Ordinary Differential Equations. Springer-Verlag, 1992.
|Title||complexification of vector space|
|Date of creation||2013-03-22 15:24:33|
|Last modified on||2013-03-22 15:24:33|
|Last modified by||stevecheng (10074)|