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

For any primitive matrix M∈Rn×n with positive diagonal entries, we prove the existence and uniqueness of a positive vector x=(x1,…,xn)t such that Mx=(1x1,…,1xn)t. The contribution of this note is to provide an alternative proof of a result of Brualdi et al. (1966) [1] on the diagonal equivalence of a nonnegative matrix to a stochastic matrix.