Gershgorin disks for multiple eigenvalues of non-negative matrices
  Gershgorin's famous circle theorem states that all eigenvalues of a square matrix lie in disks (called Gershgorin disks) around the diagonal elements. Here we show that if the matrix entries are non-negative and an eigenvalue has geometric multiplicity at least two, then this eigenvalue lies in a smaller disk. The proof uses geometric rearrangement inequalities on sums of higher dimensional real vectors which is another new result of this paper. 
Imre B\'ar\'any and J\'ozsef Solymosi
math.CO math.SP