Core-Nilpotent Decomposition

Definition
If $$\mathbf A$$ is an $$n\times n$$ matrix such that $$\text{rank} (\mathbf A^k) = r$$, then there exists a nonsungular matrix $$\mathbf Q$$ such that

$$\mathbf Q^{-1} \mathbf{AQ} = \begin{pmatrix} \mathbf C_{r\times r} & \mathbf 0 \\ \mathbf 0 & \mathbf N \end{pmatrix}$$

in which $$\mathbf C$$ is nonsingular and $$\mathbf N$$ is nilpotent of index $$k$$.

In other words, $$\mathbf A$$ is similar to a $$2\times 2$$ block-diagonal matrix containing a non-singular core and a nilpotent component.