Abstract

Guo[W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261-270] sets the question: if the list ? = {?1, ?2, ..., ?n} is symmetrically realizable (that is, ? is the spectrum of a symmetric nonnegative matrix), and t > 0, whether or not the list ?t = {?1 + t, ?2 ± t, ?3, ..., ?n} is also symmetrically realizable. In this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant. We also give a necessary and sufficient condition for ? to be the spectrum of a nonnegative left circulant matrix. © 2009 Elsevier Inc. All rights reserved.