The Karpelevic region revisited
Loading...
Date
2020
Authors
Kirkland, S.
Laffey, T.
Smigoc, H.
Journal Title
Journal ISSN
Volume Title
Publisher
Journal of Mathematical Analysis and Applications
Abstract
We consider the Karpelevic region \Theta_n in the complex plane consisting of all eigenvalues of all stochastic matrices of order n. We provide an alternative characterisation of \Theta_n that sharpens the original description given by Karpelevic. In particular,
for each \theta in [0; 2\pi); we identify the point on the boundary of \Theta_n with argument
\theta. We further prove that if n is a natural number with n at least 2, and t is in \Theta_n, then t is a subdominant eigenvalue of some stochastic matrix of order n.
Description
Keywords
stochastic matrix, eigenvalue, Markov chain