A characterization of trace zero bisymmetric nonnegative 5 × 5 matrices

Let λ1≥λ2≥λ3≥λ4≥λ5≥−λ1 be real numbers such that ∑i=15λi=0. In [16], O. Spector proved that a necessary and sufficient condition for λ1,λ2,λ3,λ4,λ5 to be the eigenvalues of a traceless symmetric nonnegative 5×5 matrix is “λ2+λ5≤0 and ∑i=15λi3≥0”. In this article, we show that this condition is also a necessary and sufficient condition for λ1,λ2,λ3,λ4,λ5 to be the spectrum of a traceless bisymmetric nonnegative 5×5 matrix.