Diagonal realizability in the Nonnegative Inverse Eigenvalue Problem

We show that if a list of nonzero complex numbers $\sigma =({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k)$ is the nonzero spectrum of a diagonalizable nonnegative matrix, then $\sigma$ is the nonzero spectrum of a diagonalizable nonnegative matrix of order $k+k^2\mathrm{.}$