A note on Guo's perturbation result for symmetric nonnegative matrices

Abstract

In 1997 Guo Wuwen proved that if $({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)$ is the spectrum of a nonnegative matrix with Perron eigenvalue ${\lambda }_1$ and ${\lambda }_2$ real, then $({\lambda }_1+t,{\lambda }_2\pm t,{\lambda }_3,\dots ,{\lambda }_n)$ is the spectrum of a nonnegative matrix for every $t\ge 0$. Whether the same holds for symmetric nonnegative matrices has remained an open question. We give an affirmative answer, in a slightly stronger form: if $A$ is a symmetric nonnegative matrix with eigenvalues ${\lambda }_1\ge \cdots \ge {\lambda }_n$ and $d_1\ge |d_2|+\cdots +|d_n|$, then there is a symmetric matrix $B\ge A$ with eigenvalues ${\lambda }_1+d_1,\dots ,{\lambda }_n+d_n$, and $B$ has the same diagonal as $A$ when $d_1+\cdots +d_n=0$. This leads us to call a symmetrically realizable list Guo-extreme if none of Guo's moves can be reversed within the set of symmetrically realizable lists. Every symmetrically realizable list is a Guo-extreme list plus a vector from the cone of Guo's moves. Our second main result is a certificate, modelled on a theorem of Laffey on extreme symmetric matrices: if $A$ is irreducible and every nonnegative symmetric matrix that commutes with $A$ and vanishes wherever $A$ is positive is a multiple of the identity, then the spectrum of $A$ can be moved by a small amount in every trace-preserving direction (in every direction, if $A$ has a positive diagonal entry) and remain symmetrically realizable; in particular it is not Guo-extreme.