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

For any primitive matrix $Minmathbb{R}^{ntimes n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $mathbf{x}=(x_1,dots,x_n)^t$ such that $Mmathbf{x}=(frac{1}{x_1},dots,frac{1}{x_n})^t$. 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.