Geometric Multiplicity

Definition

$Geo(\lambda )$ of an Eigenvalue $\lambda$ is the dimension $dim(E_{\lambda })$ for Eigenspace of $\lambda$

Example

For the matrix

$$\left[\begin{array}{cccc}1& 1& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 3\end{array}\right]$$

We get that the Eigenvalues are: ${\lambda }_1=1,{\lambda }_2=2,{\lambda }_3=3$ Finding $Geo(1)$:

• $E_1=ker(T-1I)$

$$\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 2& 0\end{array}\right]\stackrel{EF}{\to }\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$$

Then, $Geo(1)=dim(E_1)=1$