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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?

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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.)

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  • $\begingroup$ But $B$ - inverse positive implies $B^{-1}x\ge 0$ for $x\ge 0$ while we need strick inequality. I'd guess that $B^{-1}$ should be irreducible or something like this to ensure semipositivity, no? $\endgroup$ – Dmitry Oct 15 '17 at 11:46
  • $\begingroup$ @Dmitry: $B^{-1}$ must have at least one positive element in each row; otherwise it'd be singular. So $B^{-1}\geq 0$ is enough to ensure that $B^{-1}x>0$ whenever $x>0.$ $\endgroup$ – Dap Oct 15 '17 at 11:48
  • $\begingroup$ Indeed! Thank you. I cannot award you the bounty yet, but will do so asap. $\endgroup$ – Dmitry Oct 15 '17 at 12:02

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