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

Abstract

Let λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 ≥ λ 5 ≥ −λ 1 be real numbers such that 5 i=1 λ i = 0. In [14], O. Spector prove that a necessary and sufficient condition for λ 1 , λ 2 , λ 3 , λ 4 , λ 5 to be the eigenvalues of a traceless symmetric nonnegative 5 × 5 matrix is "λ 2 + λ 5 < 0 and 5 i=1 λ 3 i ≥ 0". In this article, we show that this condition is also a necessary and sufficient condition for λ 1 , λ 2 , λ 3 , λ 4 , λ 5 to be the spectrum of a traceless bisymmetric nonnegative 5 × 5 matrix.2010 Mathematics Subject Classification. 15A18.