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Let $A$ be a nonnegative symmetric $ 5 \times 5 $ matrix with eigenvalues $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq \lambda_5 $. Weshow that if $ \sum_{i=1}^{5} \lambda_{i} \geq \frac{1}{2} \lambda_1 $ then $\lambda_3 \leq \sum_{i=1}^{5} \lambda_{i} $. McDonald and Neumann showed that $\lambda_1 + \lambda_3 + \lambda_4 \geq 0 $. Let $ \sigma = \left( \lambda_1,\lambda_2, \lambda_3, \lambda_4, \lambda_5 \right) $ be a list of decreasingreal numbers satisfying:  1. $ \sum_{i=1}^{5} \lambda_{i} \geq \frac{1}{2} \lambda_1 $,  2. $ \lambda_3 \leq \sum_{i=1}^{5} \lambda_{i} $,  3. $ \lambda_1 + \lambda_3 + \lambda_4 \geq 0 $,  4. the Perron property, that is $ \lambda_1 = \max_{\lambda \in \sigma}\left| \lambda \right| $.  We show that $ \sigma $ is the spectrum of a nonnegative symmetric $ 5 \times5 $ matrix. Thus, we solve the symmetric nonnegative inverse eigenvalue problemfor $ n = 5 $ in a region for which a solution has not been known before.

A Necessary Condition for the Spectrum of Nonnegative Symmetric $ 5 \times 5 $ Matrices