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.

a Frobenius multiset if the following conditions hold: The Frobenius theorem for irreducible A 2 R nÂn þ , i.e. (I þ A) nÀ1 is a positive matrix, claims that (Ã(A)) > 0 and Ã(A) is a Frobenius set. In particular, an We say that a multiset Similarly, we extend the results of Kim et al. [5] to a nonzero eigenvalue multiset of nonnegative irreducible matrices with integer entries.

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Then A is irreducible and the nonzero part of eigenvalue multiset Ã(A) is Ã. g

An extension of Kim-Ormes-Roush theorem
In this section we give necessary and sufficient conditions on a multiset Ã of nonzero complex number to be a nonzero eigenvalue multiset of a nonnegative irreducible matrix with integer entries. Recall the Mo¨bius function : N ! {À1, 0, 1}. First, (1) ¼ 1. Assume that n > 1. If n is not square free, i.e. n is divisible by l 2 for some positive integer l > 1, then (n) ¼ 0. If n > 1 is square free, let !(n) be the number of distinct primes that divide n. Then (n) ¼ (À1) !(n) . The following theorem is a generalization of the Kim-Ormes-Roush theorem [5].
THEOREM 3.1 Let Ã be a multiset of nonzero complex numbers. Then Ã is a nonzero eigenvalue multiset of a nonnegative irreducible matrix with integer entries if and only if the following conditions hold: (1) Ã is a Frobenius set.
(3) t k ðÃÞ : Proof Assume that Ã is a nonzero spectrum of a nonnegative irreducible matrix with integer entries, i.e. A 2 Z NÂN þ . Then part 1 follows from the Frobenius theorem. Since det(zI À A) has integer coefficients, we deduce part 2. It is known that t k (Ã) ¼ t k (Ã(A)) is the number of minimal loops of length k in the directed multigraph induced by A (see [1]). Hence part 3 holds.