We learned in linear algebra that an $N \times N$ matrices can be placed into jordan normal forms with "blocks": $$ \left[ \begin{array}{cccc} \lambda & 1 & 0 & 0 \\\\ 0 & \lambda & 1 & 0 \\\\ 0 & 0 & \lambda & 1 \\\\ 0 & 0 & 0 & \lambda \end{array} \right] \hspace{0.25in}\text{or }\hspace{0.25in} \left[ \begin{array}{cccc} \lambda & 0 & 0 & 0 \\\\ 0 & \lambda & 0 & 0 \\\\ 0 & 0 & \lambda & 0 \\\\ 0 & 0 & 0 & \lambda \end{array} \right]$$

What is the analogue of "blocks" for a pair of $N \times N$ matrices, $\phi,\psi:V \to V$ ? and for a triple of matrices. $\phi, \psi,\rho: V \to V $ ?