May 7, 2010 DETERMINING WHETHER A MATRIX IS STRONGLY EVENTUALLY NONNEGATIVE

A matrix A can be tested to determine whether it is eventually positive by ex1 amination of its Perron-Frobenius structure, i.e., by computing its eigenvalues and left and right 2 eigenvectors for the spectral radius ρ(A). No such “if and only if” test using Perron-Frobenius prop3 erties exists for eventually nonnegative matrices. The concept of a strongly eventually nonnegative 4 matrix was was introduced in [2] to define a class of eventually nonnegative matrices with a stronger 5 connection to Perron-Frobenius theory (and to exclude nilpotent matrices and related problems). 6 This paper presents an algorithm that uses necessary and sufficient Perron-Frobenius-type condi7 tions to determine whether a matrix is strongly eventually nonnegative. To establish the validity of 8 the algorithm, eventually r-cyclic matrices are defined, and it is shown that a strongly eventually 9 nonnegative matrix that is not eventually positive is eventually r-cyclic, and an eventually r-cyclic 10 matrix A having rankA2 = rankA is r-cyclic. 11