Abstract

In this paper we present a sufficient condition and necesary condition for SYmmetric Nonnegative Inverse Eigenvalue Problem. This condition is independent of the existing realizability criteria. This criterion is recursive, that is, it determines whether a list $\Lambda=\{\lambda_1,....,\lambda_n,\lambda_{n+1}\}$ is realizable by a nonnegative symmetric matrix, if the list $\mu=\{\mu_1,...,\mu_n\}$ associated to $\Lambda$ is realizable. This result is easy to program and improves some existing criteria.