Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
A D1207: Complex number $\lambda \in \mathbb{C}$ is an eigenvalue for $A$ if and only if
\begin{equation}
\exists \, z \in \mathbb{C}^{N \times 1} \setminus \{ \boldsymbol{0} \} :
A z = \lambda z
\end{equation}