As in the title, I am interested in conditions on an $n\times n$ matrix $B$, such that $$(I-B)^{-1}\quad\text{is a}\ Z-\text{matrix}$$ i.e. a matrix with nonpositive off-diagonal elements. The context where the problem arises is that of stability of dynamical systems
Actually, I am interested in such a result under the following more restrictive assumptions.:
- $det(I-B)>0$ (or even the stronger: All eigenvalues of $I-B$ have positive real part)
- $-B$ with nonnegative off-diagonal entries and zeros on diagonal, implying $I-B$ with nonnegative off-diagonal entries and ones on diagonal. In particular $I-B$ is nonnegative.
My naive approach consists in observing that, thanks to the determinant condition, the off-diagonal cofactors of $I-B$ have the same sign of the off-diagonal elements of $(I-B)^{-1}$. Hence one can reduce to the following question:
$\bullet$ Given $-B$ as in the second point above, under which conditions $I-B$ has nonpositive off-diagonal cofactors?