Given $A$ in $R^{nxn}$ and its regular splitting M and N (A = M - N), $M$ is nonsingular and $M^{-1}$ and $N$ are nonnegative.
If the spectral radius $p(M^{-1}N)<1$, show $A$ nonsingular and $A^{-1}$ nonnegative.
What I've got: $A = M(I - M^{-1}N)$, but lost from there.
If $A$ is nonsingular and $A^{-1}$ is nonnegative, show that $p(M^{-1}N)<1.$
What I've got: I want to use the Perron-Frobenius theory on $M^{-1}N$, but not really sure how to do so.
From $\rho(M^{-1}N)<1$, in particular, the fact that $1$ is not an eigenvalue of $M^{-1}N$, and $M^{-1}$ is nonsingular, we have that $A=M(I-M^{-1}N)$ is nonsingular. Next, $A^{-1}=(I-M^{-1}N)^{-1}M^{-1}$. Since $\rho(M^{-1}N)<1$, $(I-M^{-1}N)^{-1}$ can be expressed in terms of Neumann series $(I-M^{-1}N)^{-1}=\sum_{i=0}^{\infty}(M^{-1}N)^i$, which shows that $(I-M^{-1}N)^{-1}\geq 0$ because each term is nonnegative. In addition, $M^{-1}\geq 0$ by definition, so $A^{-1}=(I-M^{-1}N)^{-1}M^{-1}\geq 0$.