  A Toeplitz matrix is one in which the matrix elements are constant along
diagonals. The Fisher-Hartwig matrices are much-studied singular matrices in
the Toeplitz family. The matrices are defined for all orders, $N$. They are
parametrized by two constants, $\alpha$ and $\beta$. Their spectrum of
eigenvalues has a simple asymptotic form in the limit as $N$ goes to infinity.
Here we study the structure of their eigenvalues and eigenvectors in this
limiting case. We specialize to the case $0<\alpha<|\beta|<1$, where the
behavior is particularly simple.
