A characterization of trace zero bisymmetric nonnegative $5\times 5$ matrices

Abstract

Let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3\ge {\lambda }_4\ge {\lambda }_5\ge -{\lambda }_1$ be real numbers such that ${\sum }_{i=1}^5{\lambda }_i=0$. In \cite{oren}, O. Spector prove that a necessary and sufficient condition for ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$ to be the eigenvalues of a symmetric nonnegative $5\times 5$ matrix is "${\lambda }_2+{\lambda }_5<0$ and ${\sum }_{i=1}^5{\lambda }_i^3\ge 0"$. In this article, we show that this condition is also a necessary and sufficient condition for ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5$ to be the spectrum of a traceless bisymmetric nonnegative $5\times 5$ matrix.