Consider the matrix $$M_1=\left[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right]$$ Quite obviousely, $$f(n)=M_1^n \left[\begin{matrix} a \\ b \end{matrix}\right] = \left[\begin{matrix} a+n b \\ b \end{matrix}\right]$$ exhibits polynomial (linear) growth. This is thanks to the non-diagonalizability of $M_1$.
Now consider $$M_k=\left[\begin{matrix} 1 & 1 \\ \exp(-k) & 1 \end{matrix}\right]$$ $$g(n)=\prod_{k=1}^n M_k \left[\begin{matrix} a \\ b \end{matrix}\right]$$ While I do not have an explicit expression for $g(n)$, I am confident it is asymptotically linear.
A third example $$M_k=\left[\begin{matrix} 1 & 1 & 0 \\ \exp(-k) & 1 & 0 \\ 0 & 0 & k \end{matrix}\right]$$ $$g(n)=\prod_{k=1}^n M_k \left[\begin{matrix} a \\ b \\ 0 \end{matrix}\right]$$ $$h(n)=\prod_{k=1}^n M_k \left[\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}\right]$$ $g(n)$ is asymptotically linear, but $h(n)$ grows superpolynomially.
Question I have a $3\times 3$ matrix $M_k$, and I have reason to believe that it exhibits asymptotically linear growth on some subspace (like $g(n)$) and superpolynomial growth (like $h(n)$) on another subspace. I seek a procedure to find the subspace in which it exhibits asymptotically linear growth.