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 $ \lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq \lambda_5 $. We show 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 decreasing real 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 \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. 
Raphael Loewy and Oren Spector
math.RA