A note on the nonzero spectra of irreducible matrices

Abstract

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.

Keywords:

• nonnegative inverse eigenvalue problem
• primitive matrices
• Boyle–Handelman conditions
• Kim–Ormes–Roush conditions

AMS Subject Classifications::

• 15A18
• 15A29
• 15A42
• 15B36
• 15B48