Question

This question is about maxplus algebra. There exists a generalized eigenmode analysis for a reducible maxplus matrix $M$. Is there a similar analysis for a product of a set of reducible matrices? This case happens in recurrence relations where the reducible matrix switches cyclically among a set of possible matrices, depending on the mode of the system: I.e $x(k+1)=A(k)x(k)$ such that $A(k)$ is an element of a finite set of matrices $M1$, $M2$, ..., $Mn$. Since the mode switch is cyclic, one can write $x(k+1)=M x(k-n)$ where $M=M_n M_{n-1} \dotsm M_2 M_1$. I searched for similar works but I only found solutions for irreducible matrices.