ELA, Volume 12, pp. 73-76, December 2005, abstract.

Perturbing Non-real Eigenvalues of Nonnegative Real Matrices

Thomas J. Laffey

Let A be an (entrywise) nonnegative nxn matrix with spectrum 
\sigma and Perron eigenvalue \rho. Guo Wuwen [Linear Algebra 
and its Applications 266 (1997), pp. 261-267] has shown that 
if \lambda is another real eigenvalue of A, then, for all t>=0, 
replacing \rho, \lambda in \sigma by \rho+t, \lambda-t, respectively, 
while keeping all other entries of \sigma unchanged, again yields 
the spectrum of a nonnegative matrix. He poses the question of 
whether an analogous result holds in the case of non-real \lambda. 
In this paper, it is shown that this question has an affirmative 
answer.