Let $A \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix.
Let $I_N \in \mathbb{C}^{N \times N}$ be a D5699: Complex identity matrix.
A D5975: Euclidean complex number $(\lambda_1, \ldots, \lambda_N) \in \mathbb{C}^N$ is an eigenvalue sequence for $A$ if and only if \begin{equation} \forall \, z \in \mathbb{C} : \text{Det}(z I_N - A) = \prod_{n = 1}^N (z - \lambda_n) \end{equation}