  This note starts from work done by Dai, Geary, and Kadanoff (Hui Dai, Zachary
Geary, and Leo P. Kadanoff, H. Dai, Z. Geary and L. P. Kadanoff, Journal of
Statistical Mechanics, P05012 (2009)) on exact eigenfunctions for Toeplitz
operators. It builds methods for finding convergent expansions for eigenvectors
and eigenvalues of large-n Toeplitz matrices, using the infinite-n case as a
starting point. One expansion is derived from operator equations having a
two-dimensional continuous spectrum of eigenvalues, which include the
eigenvalues of the finite-$n$ matrices. Another expansion is derived from the
transpose equations, which have no eigenvalues at all. The two expansions work
together to give an apparently convergent expansion with an expansion parameter
expressed as an inverse power of n. A variational principle is developed which
gives an approximate expression for determining eigenvalues. A consistency
condition is generated, which gives to lowest order exactly the same condition
for the eigenvalue.
