Let be a vector space over a field and a linear transformation on (a linear operator). A non-zero vector is said to be a generalized eigenvector of (corresponding to ) if there is a and a positive integer such that
where is the identity operator.
In the equation above, it is easy to see that is an eigenvalue of . Suppose that is the least such integer satisfying the above equation. If , then is an eigenvalue of . If , let . Then (since ) and , so is again an eigenvalue of .
Let be a generalized eigenvector of corresponding to the eigenvalue . We can form a sequence
The set of all non-zero terms in the sequence is called a cycle of generalized eigenvectors of corresponding to . The cardinality of is its . For any , write .
Below are some properties of :
is the only eigenvector of in , for otherwise .
Let , where . Let with . Induct on . If , then , so and is linearly independent. Suppose the property is true when . Apply to the equation, and we have . Then by induction. So and thus since is an eigenvector and is non-zero. ∎
More generally, it can be shown that is linearly independent whenever is.
A cycle of generalized eigenvectors is called maximal if . If is finite dimensional, any cycle of generalized eigenvectors can always be extended to a maximal cycle of generalized eigenvectors , meaning that .
In particular, any eigenvector of can be extended to a maximal cycle of generalized eigenvectors. Any two maximal cycles of generalized eigenvectors extending span the same subspace of .
- 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
|Date of creation||2013-03-22 17:23:13|
|Last modified on||2013-03-22 17:23:13|
|Last modified by||CWoo (3771)|
|Defines||cycle of generalized eigenvectors|