# generalized eigenvector

Let $V$ be a vector space  over a field $k$ and $T$ a linear transformation on $V$ (a linear operator). A non-zero vector $v\in V$ is said to be a generalized eigenvector  of $T$ (corresponding to $\lambda$) if there is a $\lambda\in k$ and a positive integer $m$ such that

 $(T-\lambda I)^{m}(v)=0,$

where $I$ is the identity operator  .

In the equation above, it is easy to see that $\lambda$ is an eigenvalue     of $T$. Suppose that $m$ is the least such integer satisfying the above equation. If $m=1$, then $\lambda$ is an eigenvalue of $T$. If $m>1$, let $w=(T-\lambda I)^{m-1}(v)$. Then $w\neq 0$ (since $v\neq 0$) and $(T-\lambda I)(w)=0$, so $\lambda$ is again an eigenvalue of $T$.

Let $v$ be a generalized eigenvector of $T$ corresponding to the eigenvalue $\lambda$. We can form a sequence

 $v,(T-\lambda I)(v),(T-\lambda I)^{2}(v),\ldots,(T-\lambda I)^{i}(v),\ldots,(T-% \lambda I)^{m}(v)=0,0,\ldots$

The set $C_{\lambda}(v)$ of all non-zero terms in the sequence is called a cycle of generalized eigenvectors of $T$ corresponding to $\lambda$. The cardinality $m$ of $C_{\lambda}(v)$ is its . For any $C_{\lambda}(v)$, write $v_{\lambda}=(T-\lambda I)^{m-1}(v)$.

Below are some properties of $C_{\lambda}(v)$:

## References

• 1 Friedberg, Insell, Spence. Linear Algebra. Prentice-Hall Inc., 1997.
Title generalized eigenvector GeneralizedEigenvector 2013-03-22 17:23:13 2013-03-22 17:23:13 CWoo (3771) CWoo (3771) 13 CWoo (3771) Definition msc 65F15 msc 65-00 msc 15A18 msc 15-00 GeneralizedEigenspace cycle of generalized eigenvectors