A note on matrices mapping a positive vector onto its element-wise inverse

For any primitive matrix $M\in {\mathbb{R}}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots ,x_n{)}^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots ,\frac{1}{x_n}{)}^t$. The contribution of this note is to provide an alternative proof of a result of Brualdi et al. (1966) on the diagonal equivalence of a nonnegative matrix to a stochastic matrix.