Journal or Book Title
Electronic Journal of Linear Algebra
Eventually r-cyclic matrices are deﬁned, and it is shown that if A is an eventually r-cyclic matrix A having rank A2 = rank A, then A is r-cyclic with the same cyclic structure. This result and known Perron-Frobenius theory of eventually nonnegative matrices are used to establish an algorithm to determine whether a matrix is strongly eventually nonnegative (i.e., is an eventually nonnegative matrix having a power that is both irreducible and nonnegative).
Hogben, Leslie, "Eventually cyclic matrices and a test for strong eventual nonnegativity" (2010). Mathematics Publications. 248.