# Definition:Eigenvector

## Definition

Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$.

Let $A \in \map B H$ be a bounded linear operator.

Let $\alpha \in \Bbb F$ be an eigenvalue of $A$.

A nonzero vector $h \in H$ is said to be an eigenvector for $\alpha$ if and only if:

$h \in \map \ker {A - \alpha I}$

That is, if and only if $Ah = \alpha h$.

### Eigenspace

The eigenspace for an eigenvalue $\alpha$ is the set $\map \ker {A - \alpha I}$.