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 parameterized by two constants, α and β. 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 with real α and β and 0<α<|β|<1, where the behavior is particularly simple.