The diagonalizable nonnegative inverse eigenvalue problem

In this paper we prove that the SNIEP $\neq$ DNIEP, i.e. the symmetric and diagonalizable nonnegative inverse eigenvalue problems are different. We also show that the minimum $t>0$ for which $(3+t,3-t,-2,-2,-2)$ is realizable by a diagonalizable matrix is $t=1$, and we distinguish diagonalizably realziable lists from general realizable lists using the Jordan Normal Form