In the background of this question is a matrix $A$, all of whose elements are positive. The Perron-Frobenius theorem tells us that the eigenvalue with largest absolute value is real, and that there is an associated *dominant eigenvector*, all of whose elements are positive.

Suppose we don't actually observe $A$, but are told what its first row sum is. We're also told the first row sum of $A^{2}$, $A^{3}$, $A^{4}$, ... . In other words, writing $e$ for the vector of ones, we're told the first element of $Ae$, $A^{2}e$, $A^{3}e$, and so on. If, for example, $A$ is a stochastic matrix then $Ae = e$ so that the information given is simply a list of ones: $(Ae)_{1}=1$, $(A^{2}e)_{1}=1$, etc.

This information is enough to work out the dominant eigenvalue of $A$ via the power method: simply compute $\lim_{n \to \infty} \left( (A^{n} e)_{1} \right)^{1/n}$.

My question is:

Can anything at all be said about the dominant eigenvector?