Note on nonnegative matrices

Let $A$ be a nonnegative square matrix and $B = {D_1}A{D_2}$ where ${D_1}$ and ${D_2}$ are diagonal matrices with positive diagonal entries. Several proofs are known for the following theorem: If $A$ is fully indecomposable then ${D_1}$ and ${D_2}$ can be chosen so that $B$ is doubly stochastic. Moreover, ${D_1}$ and ${D_2}$ are unique up to a scalar factor. It is shown that these results can be easily obtained by considering a minimum of a certain rational function of several variables.