# Sufficient conditions for inverse-positivity

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?

• 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. Apr 25, 2012 at 10:20

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.
$\mathbb{R}^n_+ \subseteq A(\mathbb{R}^n_+)$ is one sufficient condition for inverse positivity.
• 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. Oct 6, 2012 at 4:10