If $A$ is non negative and has a positive eigenvector $ \Rightarrow $ A is diagonally similar to a non negative matrix If $A\in M_n$ is non negative(all $a_{ij}\ge 0$), and has a positive eigenvector(all $x_i>0$), why is $ A$ diagonally similar to a nonnegative matrix, all of whose row sums are equal?

**Hint.** If $v$ is a vector, then $Av\equiv A\operatorname{diag}(v)\mathbf1$, where $\operatorname{diag}(v)$ is the diagonal matrix whose diagonal entries are elements of $v$ and $\mathbf 1$ is the vector of ones.