Relation between operator norm and the minimum component of Perron-Frobenius eigenvector

Question

Let $A$ be an $n \times n$ matrix with positive entries. Is it possible to find any relation between its operator norm and the minimum component of its Perron-Frobenius eigenvector (component with minimum value)?

Answer 1

I presume that we normalize the Perron vector so it has unit norm (say in the uniform norm)? I also presume that we normalize $A$ with respect to its Perron value $\lambda$?

Take $$A=\left( \begin{matrix} 1 & 1/N \\ 1/N & 1 \end{matrix} \right)$$

Then $\lambda=1+1/N$ and $v=\left( \begin{matrix} 1 \\ 1 \end{matrix} \right)$. Operator norm goes to one, but the minimum value of the matrix element goes to zero as $N\rightarrow \infty$. Replacing the lower right element in $A$ by $1/N$ the minimum component of the Perron vector goes to zero (without affecting the rest), so you will need to put more restrictions on your matrices.

There are results (fairly easy) if you assume that $\min A_{ij} / \max A_{kl}$ is bounded from below by some constant. Perhaps of interest?