Let $\Sigma$ be a finite non-empty set of symbols (i.e. an alphabet). Fix $\pi, \eta\in\mathbb{R}^{1\times m}$ and for every $\sigma\in\Sigma $ fix $A(\sigma)\in\mathbb{R}^{m\times m}$. We also require that for every $1 \leq i,j\leq m$ $\pi_i, \eta_i, (A(\sigma))_{i,j} \geq 0$ but $\pi,\eta,A(\sigma)\neq 0$.

If $w:=w_1\cdots w_n\in\Sigma^*$ let $A(w) := \prod_{i=1}^{n} A(w_i)$. Suppose that there exists a real $K$ such that for every $w\in\Sigma^*$ one has $|\pi A(w)\eta^{t} |\leq K$.

Is it true that this imply the existence of $K'$ such that for every $w\in\Sigma^*$ $||\pi A(w)||\leq K'$ for some $K'$? If not, any counterexample?

Here $||\cdot||$ denotes the usual $\ell_2$ norm of a real vector.