Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is actually a spiced-up restatement of my older question.
$\begingroup$
$\endgroup$
3
asked Jul 3, 2012 at 14:24
-
1$\begingroup$ How is the Birkhoff norm of a vector defined? $\endgroup$– Lee MosherCommented Jul 3, 2012 at 14:37
-
$\begingroup$ If I understand correctly, the Perron–Frobenius theorem asserts that the unique largest eigenvalue of $A$ has as its corresponding eigenvector a vector $x$ with entirely positive entries ($x$ is determined up to non-zero real scalar), and you are asking for the ratio of the largest to least entry of $x$? Anyway, I don't see how this is different from the older linked question. $\endgroup$– Theo Johnson-FreydCommented Aug 27, 2013 at 6:58
-
$\begingroup$ @TheoJohnson-Freyd It's different in the way I was approaching the subject the first and the second time. The first time I was thinking in elementary terms and the second time I realized that perhaps the question could be better addressed by couching it in more general language. Whether this in retrospect merited a separate question or a major edit I am not sure, but for me it felt like a practically new one in real time. $\endgroup$– Felix GoldbergCommented Aug 27, 2013 at 7:08
Add a comment
|