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

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Mar 25, 2012 at 17:04	vote	accept	Ian Martin
Mar 25, 2012 at 2:45	comment	added	Chris Godsil		@Ian. An $n\times n$ matrix $A$ can be reconstructed from the vectors $x$,\dots,$A^{n-1}x$ if and only if these vectors span $\mathbb{R}^n$. But this is using $n$ times as much information. Roughly speaking, in your case you can reconstruct one entry of each eigenvector. Fedja's answer provides another way of viewing this point.
Mar 25, 2012 at 1:57	comment	added	Ian Martin		Thanks @Chris, very nice! Certainly this answers the implied question in my title. I wonder whether there's anything at all (beyond Perron-Frobenius) that can be said about the dominant eigenvector, though? If I'm not mistaken, knowledge of the entries of the first row for $A$, $A^{2}$, $A^{3}$, etc, is enough to reconstruct the whole matrix (at least so long as $A$ is not unipotent). So I was hoping that knowledge of the row sums might at least provide some information about the dominant eigenvector. But perhaps that's too much to hope for.
Mar 24, 2012 at 23:08	history	answered	Chris Godsil	CC BY-SA 3.0