Geometric Multiplicity of Eigenvalues

[danielpanatha]

Could someone please explain to me (with an example if possible) what is the Geometric Multiplicity of Eigenvalues? I cannot understand it from what I have read on the web till now.
Thanks in advance.

 

[micromass]

Let $\lambda$ be an eigenvalue of the linear transformation $T:V\rightarrow V$.
We can look at the set of all eigenvectors (plus 0). So we get

$$E_\lambda = \{v\in V~\vert~Tv = \lambda v\}$$

This is a subspace of $V$ and thus it has a dimension. The dimension of $E_\lambda$ is called the geometric multiplicity of $\lambda$.