Let $\text{Diag}(\mathbb{C}^{N \times N})$ be a D5993: Set of diagonal complex matrices.
Let $\text{Inv}(\mathbb{C}^{N \times N})$ be a D6219: Set of invertible complex matrices.
A D6159: Complex square matrix $A \in \mathbb{C}^{N \times N}$ is a diagonalizable complex matrix if and only if \begin{equation} \exists \, P \in \text{Inv}(\mathbb{C}^{N \times N}) : P^{-1} A P \in \text{Diag}(\mathbb{C}^{N \times N}) \end{equation}