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Suppose you have a real symmetric matrix $A$ which is positive except for $a_{ij},a_{ji}$, who are negative.

While it is not generally true that the eigenvector of the dominant eigenvalue of $A$ is positive (I have a 7x7 counterexample which I can post if somebody cares for it), I wonder if under some additional assumptions (maybe on the value of the negative entries), this eigenvector can be proved to be positive?

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  • $\begingroup$ Symmetric matrix is easy case. Cause you know eigval is real. Some sufficient conditions I think easy to get something like diagonal domination.... but it can be very far from necessary ..... $\endgroup$
    – Alexander Chervov
    Commented Mar 31, 2012 at 7:46
  • $\begingroup$ I'm not sure about the diagonal dominance - it can ensure positive eigenvalues, but it's the eigenvector I'm after. Or am I missing something? $\endgroup$
    – Felix Goldberg
    Commented Mar 31, 2012 at 11:27
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    $\begingroup$ @Felix Yes, you are right, it was not quite correct. Still I think the following makes sense - pertrubation theory argument. If A= Positive + Epsilon. We can estimate how the eigenvectors are changed - if Epsilon is sufficiently small we can hope for positivity. Still here is some gap - I do not know how to ensure that all elements of eigenvector are big enough. Except the only case - stochastic matrices have PF-eigenvector = (1,1,1,1,...,1). So in this case I think it can be gained something from pertrubation theory. $\endgroup$
    – Alexander Chervov
    Commented Apr 8, 2012 at 10:06

2 Answers 2

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The following paper (and the large number of references cited therein) provides some general sufficient conditions to ensure the "Perron-Frobenius property," thereby offering a set of useful answers to your question.

Reference

  1. A. Elhashash, D. B. Szyld. On general matrices having the Perron-Frobenius property. Electronic J. Linear Algebra, vol. 17, pp. 389--413. (2008).
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  • $\begingroup$ Thanks! I am familiar with this one actually, in broad terms: it links Perron-Frobeniusness to eventual positivity. I was hoping to come across some other kind of condition (like the Tarazaga-Horn condition) that I could apply. In the end I used the extra special information about my matrix to prove what I needed. But thanks anyway! $\endgroup$
    – Felix Goldberg
    Commented Apr 7, 2012 at 9:46
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I think I found something! After long and hard googling, I stumbled upon this:

MR2311036 (2008e:15011) 15A18 (15A29 15A48) Chen, Jianbiao; Xu, Zhaoliang The inverse eigenvalue problem for real eventually positive matrices. Filomat 21 (2007), no. 1, 1–16.

http://www.doiserbia.nb.rs/img/doi/0354-5180/2007/0354-51800701001C.pdf

The result that interests me is a structure theorem for eventually positive matrices (Theorem 3.2):

A real $n \times n$ matrix $A$ is eventually positive iff there exist positive vectors $\alpha,\beta$ (of length $n$) and a $n \times n$ matrix $Y$ so that:

$A=\frac{1}{(\beta^{T}\alpha)^{2})}\alpha\beta^{T}+\frac{1}{\beta^{T}\alpha}(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})Y(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})$ and the spectral radius of $(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})Y(I-\frac{1}{\beta^{T}\alpha}\alpha\beta^{T})$ is $<1$.

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