Suppose I have the following companion matrix ($d\times d$)
[The companion matrix A](https://i.stack.imgur.com/KgkoY.jpg). $1 \geq p \geq q \geq 0$. Let $x$ ($d\times 1$) be the all one vector and my underlying problem is to analyze the first entry of $A^nx$ for some large $n$. Even if the close form doesn't exist, we know that the behavior of it will determined by the largest eigenvalue of matrix $A$.
So here's my question: Is it possible to find the nice bound/approximation of the largest eigenvalue of the matrix $A$ (in terms of $d,p,q$)? Any direction or reference would be appreciated!



Bounding/approximating the largest eigenvalue of the special case of companion matrix
First of all, we can scale your matrix so there is only one parameter, so let's say $p = 1$ for simplicity.


It looks to me like the characteristic polynomial of your matrix is
$$ P(\lambda) = \frac{\lambda^{n+1} - \lambda^n + q^n - q^{n+1}}{\lambda - q} $$
Thus $P(1) = q^n$ and $P(q) = (n+1) q^n - n q^{n-1}$. In particular, if $q < n/(n+1)$ we have $P(q) < 0 < P(1)$, so there is a root between $q$ and $1$.


