Calculating the Perron-Frobenius eigenvector of a positive matrix from limited information 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?

 A: Assume that $v_1,\dots, v_n$ are eigenvectors and $\lambda_1,\dots,\lambda_n$ are eigenvalues. Assume also that $e=\sum_k v_k$. Then what you are given is merely $\sum_n (v_k,e_1) \lambda_k^n$, which, in a generic position, allows you to determine $\lambda_k$ and $(v_k,e_1)$ but no more than that. Now, take any matrix $A$ with positive entries, find the eigenvectors and eigenvalues, keep the eigenvalues but change the eigenvectors a tiny bit (respecting complex conjugation) by changing the coordinates of $v_k$ beyond the first to keep the identity $\sum v_k=e$. You'll get a new matrix with real entries close to the old one so it'll still have positive entries and the same sequence of observables but rather different eigenvectors.  
A: There is some very partial information you can obtain. See this recent paper:
Das, Kinkar Ch.
A sharp upper bound on the maximal entry in the principal eigenvector of
symmetric nonnegative matrix. (English)
Linear Algebra Appl. 431, No. 8, 1340-1350 (2009). ISSN 0024-3795
