The Diagonalizable Nonnegative Inverse Eigenvalue Problem

In this article we provide some lists of real numbers which can be realized as the spectra of nonnegative diagonalizable matrices but which are not the spectra of nonnegative symmetric matrices. In particular, we examine the classical list σ = (3 + t, 3 − t, −2, −2, −2) with t ≥ 0, and show that 0 is realizable by a nonnegative diagonalizable matrix only for t ≥ 1. We also provide examples of lists which are realizable as the spectra of nonnegative matrices, but not as the spectra of nonnegative diagonalizable matrices by examining the Jordan Normal Form.