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.



On the boundedness of linear representations of formal power series of languages.
Here is a counterexample. Suppose that $A(\sigma)$ is upper
triangular with a $1$ in upper left position, and that
$\eta=[1\ 0\ 0\ \cdots\ 0]$ and $\pi=[1\ 1\ 1\cdots\ 1]$.
Note that $A(w)$ is also upper triangular with a $1$ at
upper left, and that $A(w)\eta^t=[1\ 0\ 0\ \cdots\ 0]^t$ and
so $\pi A(w)\eta^t=1$ for all $w$. But meanwhile, it is
easy to arrange that $\pi A(w)$ can become unboundedly
large by taking $w$ large.


For a concrete example, use $A(\sigma)=[1\ 1;\ 0\ 1]$ and
$\eta=[1\ 0]$ and $\pi=[1\ 1]$.


