I have a matrix in the form :

$$M = \begin{pmatrix} A & 0 & 0 \\\ B & A & 0 \\\ C & D & A \end{pmatrix} $$

where $A,B,C,D$ are diagonalizable square matrices and I want to determine

$$M^{\infty}:=\lim_{n\rightarrow \infty} M^n$$

as a function of $A,B,C,D$.

Thank you for your help !