MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and others) and can be seen here:

The nicest bound for my case seems to be $\frac{\max{x_{i}}}{\min{x{i}}} \leq \frac {R-r+m_{2}}{m_{2}}$, where $R,r$ are the largest and smallest rows sums and $m_{2}$ is smallest off-diagonal entry.

However, it is difficult for me to use this bound since I do not have an easy explicit form for the matrix. I do have the value of the Perron root.

So, I'd like to know if there has been any progress on this problem. (I am aware of a later paper by de Oliveira and a few recent papers that deal with special cases, but they appear to be inapplicable here).

share|cite|improve this question
Take two arbitrary positive column vectors $l$ and $r$ and consider the matrix $rl^t$ to see by scaling one of the vectors, that the Perron root alone will give no control of the Birkhoff norm of the Perron-Frobenius-vectors. I recently answered an essentially duplicate version of your question: <…;, but you know that material yourself. I would be glad to hear about any recent results in this direction as I might need them myself, so I upvoted this one. – thomashennecke Mar 29 '14 at 12:50

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.