On common eigenvectors for semigroups of matrices in tropical and   traditional mathematics
  We prove the existence of a common eigenvector for commutative, nilpotent and quasinilpotent semigroups of matrices with complex or real nonnegative entries both in the conventional and tropical linear algebra. 
Grigory Shpiz, Grigory Litvinov and Sergei Sergeev
math.RA