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I was studying to my research and get stopped in a proof of a question of the book "nonnegative matrices in the mathematical sciences", by Berman & Plemmons. The question 5.2 of page 159 says "Show that if A and B are $n \times n$ M-matrices and if $AB \in Z^{n \times n}$, then $AB$ is an M-matrix."

Furthermore, the autors has sayed to use lemma 4.1, that states "Let $A \in Z^{n \times n}$. Then A is a M-matrix if and only if $A+\epsilon I $ is a nonsingular M-matrix for all sclars $\epsilon>0$"

Can someone help me to solve this question? I have tried to use the lemma in $A$ and $B$ and in $AB$ to show a equivalence of the conclusion, but nothing works yet. Thanks for any collaboration.

anderstood
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Rafael Souto
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  • Clarification please: $A,B$ are $M$-matrices, so they are $\mathbb{Z}^{n\times n}$. Then necessarily $AB\in\mathbb{Z}^{n\times n}$ so I don't understand the "if $AB\in\mathbb{Z}^{n\times n}$". – anderstood Oct 16 '23 at 21:44
  • Think in these two $Z^{2\times 2}$ matrices: $A=\begin{pmatrix} -2 & -1\ 0 & 3 \end{pmatrix}$ and $B=\begin{pmatrix} 4 & -3\ -1 & -1 \end{pmatrix}$. Then $AB=\begin{pmatrix} -7 & 7\ -3 & -3 \end{pmatrix}$, that isn't a Z matrix, so make sense to do this hypotesis to explicit this property. – Rafael Souto Oct 17 '23 at 18:48
  • OK, I wrongly interpreted $Z$ matrix as a matrix over $\mathbb{Z}$. – anderstood Oct 18 '23 at 07:31

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Let $A_\epsilon:=A+\epsilon I$ and $B_\epsilon:=B+\epsilon I$. When $\epsilon>0$ is sufficiently small, $A_\epsilon$ and $B_\epsilon$ are nonsingular M-matrices and hence they are inverse-positive. Also, $A_\epsilon B_\epsilon=AB+\epsilon(A+B)+\epsilon I$ is a nonsingular Z-matrix and it is inverse-positive because $A_\epsilon$ and $B_\epsilon$ are inverse-positive. Therefore $A_\epsilon B_\epsilon$ is an M-matrix. Since the eigenvalues of a matrix are continuous functions of the matrix entries, the eigenvalues of $AB=\lim_{\epsilon\to0^+}A_\epsilon B_\epsilon$ also have nonnegative real parts. Since $AB$ is also a Z-matrix, it is an M-matrix.

user1551
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