$\begingroup$This problem comes from the book "Topics in matrix analysis" by Horn and Johnson.
Let $A$ and $B$ be two $M$-matrices. Show that $B^{-1}A$ is an $M$-matrix if and only if $B^{-1}A$ is a $Z$-matrix.
I cannot figure out which characterization of $M$-matrices should be used to prove $B^{-1}A\in\mathbf{Z}\Rightarrow B^{-1}A\in\mathbf{M}$. Could someone give a clue?
$\endgroup$1 Answer
$\begingroup$Use that $B^{-1}$ is inverse positive and $A$ is semipositive to show that $B^{-1}A$ is semipositive. (Using terminology from that Wikipedia page.)
$\endgroup$3
