Let be a vector space (over a field ), and a linear operator on , and an eigenvalue of . The set of all generalized eigenvectors of corresponding to , together with the zero vector , is called the generalized eigenspace of corresponding to . In short, the generalized eigenspace of corresponding to is the set
Here are some properties of :
, where is the eigenspace of corresponding to .
If is finite dimensional, then is the algebraic multiplicity of .
iff . More generally, iff and are disjoint sets of eigenvalues of , and (or ) is defined as the sum of all , where (or ).
If is finite dimensional and is a linear operator on such that its characteristic polynomial splits (over ), then
where is the set of all eigenvalues of .
Assume that and have the same properties as in (5). By the Jordan canonical form theorem, there exists an ordered basis of such that is a Jordan canonical form. Furthermore, if we set , then , the matrix representation of , the restriction of to , is a Jordan canonical form. In other words,
where each is a Jordan canonical form, and is a zero matrix.
Conversely, for each , there exists an ordered basis for such that is a Jordan canonical form. As a result, with linear order extending each , such that for and for , is an ordered basis for such that is a Jordan canonical form, being the direct sum of matrices .
More to come…
- 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
|Date of creation||2013-03-22 17:23:36|
|Last modified on||2013-03-22 17:23:36|
|Last modified by||CWoo (3771)|