An Equilibrium Problem for the Limiting Eigenvalue Distribution of Rational Toeplitz Matrices

We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize recent results by Kuijlaars and one of the authors [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 173–196] that were concerned with banded Toeplitz matrices.