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](https://mathoverflow.net/questions/95097/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?



Sufficient conditions for inverse-positivity
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. 


