Matrix roots of imprimitive irreducible nonnegative matrices
  Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible nonnegative matrices. Preliminary results concerning the matrix roots of reducible matrices are provided as well. 
Judith J. McDonald, Pietro Paparella
math.RA