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I have a matrix $J(x)$ with $J_{ij}(x)=f_{ij}(x)$ where vector $x$ is $x=x_1, x_2, ..., x_m$. I have shown that $J(x)$ is an M-matrix for all $x$. There is known review paper by Plemmons (1977) of 40 equivalent properties of M-matrices. One of this properties is that if $J$ (with constant entries) is an M-matrix then exists a positive diagonal matrix $D$ such that for each nonzero vector $y$ holds $y^TJDy > 0$. My question is if a matrix $J(x)$ is an M-matrix for all $x$ will this property hold? That is will there always be a constant positive diagonal matrix $D$ such that for all nonzero vectors $y$ holds $y^TJ(x)Dy > 0$?

Any help appreciated.

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