I have a (column) substochastic regular nonnegative matrix ($P$) obeying the Perron-Frobenius theorem. I am interested in the bounds on the dominant eigenvalue (Perron root). I know that I can bound the root quite well based on the maximum and minimum row sums (one of the consequences of PF theorem) which is helpful. However, is there any other (sharp) bounds that I can exploit for estimating or bounding the Perron root?
I also know the power method for getting the dominant eigenvalue and how that can give us better bounds.
I have tried working with the bounds discussed in https://www.sciencedirect.com/science/article/pii/S0024379504001685 but they are significantly worse than the PF bounds.
P.S. Similarly I am interested in bounds for the corresponding left ($u$) and right ($v$) eigenvectors. I normalize them in the following way:
$\sum_i v_i = 1$
$\sum_i u_i v_i = 1$
It is clear that $0 < v_i < 1$ and $ u_i > 0$. For a little while I thought $u_i > 1/\rho$, but eventually found a counter example.
You can get similar, and stricter, bounds for powers of the matrix. Let $\lambda$ be the Perron root of $P$.
It follows $\lambda^2$ is the Perron root for $P^2$, and therefore the maximum and minimum row sums of $P^2$ give bounds on $\lambda^2$; now take their square roots and you get bounds on $\lambda$ itself.
And you can do this for any power at all: it follows that $\lambda^n$ is the Perron root for $P^n$, and therefore the maximum and minimum row sums of $P^n$ give bounds on $\lambda^n$; now take their $n^{\text{th}}$ roots to get bounds on $\lambda$ itself.
These bounds will get better and better, in fact they will coverge to $\lambda$, as $n$ goes to $+\infty$. Not only that, but the convergence is pretty good.