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I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the question might have general interest). This matrix is not a $Z$-matrix so the whole body of knowledge that had been developed for them is not suitable.

What I'd like to find is a simple sufficient condition that I can easily analyze. The best approximation to that that I've found is Peris's criterion using $B$-splittings but it still didn't help me enough.

Are you aware of any such results?

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One example I've just found is sciencedirect.com/science/article/pii/S0024379508005752 - where just such results are obtained for small sign-changing perturbations of tridiagonal $M$-matrices. I'm looking for more general results. –  Felix Goldberg Apr 25 '12 at 10:20

2 Answers 2

up vote 1 down vote accepted

Sorry for promoting my own results, but I think the condition in my old paper "A sufficient condition for the monotonicity of a positive definite matrix" (Computational Mathematics and Mathematical Physics vol. 41, No 9., pp. 1237-1238) may be of help. (Unfortunately, I lost the file years ago). I don't know whether it actually works, for you didn't say much about your $Q$, but it looks promising.

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Is there an English version anywhere? (I can read the Russian but I like to have an English version). –  Felix Goldberg Jan 11 '13 at 14:44
    
This journal is translated, so there must be an English copy somewhere. –  Alex Gavrilov Jan 11 '13 at 15:14
    
You may try this: maik.rssi.ru/cgi-perl/journal.pl?name=commat&page=main –  Alex Gavrilov Jan 11 '13 at 15:16
    
I accepted the answer, it looks interesting! The last link you gave didn't seem to lead to a softcopy I could obtain but I've read the Russian text and it's good enough for me! I'll see now if it helps with my original problem but it's a great result anyway. –  Felix Goldberg Jan 21 '13 at 13:55
    
Thanks. Good luck. –  Alex Gavrilov Jan 22 '13 at 11:11

$\mathbb{R}^n_+ \subseteq A(\mathbb{R}^n_+)$ is one sufficient condition for inverse positivity.

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That's the definition essentially... –  Felix Goldberg Oct 5 '12 at 11:51
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What about semipositivity ? An $m \times n$ matrix is said to be semipositive if there exists $x \in \mathbb{R}^n_+$ such that $Ax \in int(\mathbb{R}^m_+)$; $A$ is said to be minimally semipositive if in addition to being semipositive, no proper $m \times p$ submatrix is semipositive. For a square matrix, minimal semipositivity is the same as inverse positivity. I do not know if MSP is easy to verify. Two references are : Johnson, Kerr and Stanford, Semipositivity of matrices, LAMA, 37(4) (1994), 265-271 and H.J Werner, Characterizations of semipositivity, LAMA, 37(4) (1994), 273-278. –  user27020 Oct 6 '12 at 4:10

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