  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.
