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\section{Eigenvalue problems}

\begin{defi}
    Let $A \in \field^{n \times n}$. Then $\lambda \in \field$ is called an eigenvalue of $A$, if
    \[
        \exists v \in \field^n, ~v \ne 0: ~~Av = \lambda v
    \]
    Such a vector $v$ is called eigenvector. We call 
    \[
        \set[Av = \lambda v]{v \in \field^n} =: E_{\lambda}
    \]
    eigenspace belonging to $\lambda$
\end{defi}
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