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

Let $A$ be a nonnegative symmetric $ 5 \times 5 $ matrix with eigenvalues $ λ_1 \geq λ_2 \geq λ_3 \geq λ_4 \geq λ_5 $. We show that if $ \sum_{i=1}^{5} λ_{i} \geq \frac{1}{2} λ_1 $ then $ λ_3 \leq \sum_{i=1}^{5} λ_{i} $. McDonald and Neumann showed that $ λ_1 + λ_3 + λ_4 \geq 0 $. Let $ σ= \left( λ_1, λ_2, λ_3, λ_4, λ_5 \right) $ be a list of decreasing real numbers satisfying: 1. $ \sum_{i=1}^{5} λ_{i} \geq \frac{1}{2} λ_1 $, 2. $ λ_3 \leq \sum_{i=1}^{5} λ_{i} $, 3. $ λ_1 + λ_3 + λ_4 \geq 0 $, 4. the Perron property, that is $ λ_1 = \max_{λ\in σ} \left| λ\right| $. We show that $ σ$ is the spectrum of a nonnegative symmetric $ 5 \times 5 $ matrix. Thus, we solve the symmetric nonnegative inverse eigenvalue problem for $ n = 5 $ in a region for which a solution has not been known before.