A note on the nonzero spectra of irreducible matrices

In this note we extend the necessary and sufficient conditions of Boyle-Handleman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249-316] and Kim-Ormes-Roush [K.H. Kim, N.S. Ormes, and F.W. Roush, The spectra of nonnegative integer matrices via formal power series, J. Am. Math. Soc. 13 (2000), pp. 773-806] for a nonzero eigenvalue multiset of primitive matrices over R+ and Z(+), respectively, to irreducible matrices.