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

Abstract

Let $A$ be a nonnegative symmetric $5\times 5$ matrix with eigenvalues ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3\ge {\lambda }_4\ge {\lambda }_5$. We show that if ${\sum }_{i=1}^5{\lambda }_i\ge \frac{1}{2}{\lambda }_1$ then ${\lambda }_3\le {\sum }_{i=1}^5{\lambda }_i$. McDonald and Neumann showed that ${\lambda }_1+{\lambda }_3+{\lambda }_4\ge 0$. Let $\sigma =\left({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5\right)$ be a list of decreasing real numbers satisfying: 1. ${\sum }_{i=1}^5{\lambda }_i\ge \frac{1}{2}{\lambda }_1$, 2. ${\lambda }_3\le {\sum }_{i=1}^5{\lambda }_i$, 3. ${\lambda }_1+{\lambda }_3+{\lambda }_4\ge 0$, 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.

61 pages