A Toeplitz matrix is one in which the matrix elements are constantalong diagonals. The Fisher–Hartwig matrices are much-studied singularmatrices in the Toeplitz family. The matrices are defined for all orders,N. They are parameterizedby two constants, α and β. Their spectrum of eigenvalues has a simple asymptotic form in the limit asN goes to infinity. Here we study the structure of their eigenvalues andeigenvectors in this limiting case. We specialize to the case with realα andβ and0<α<|β|<1, where the behavior is particularly simple.The eigenvalues are labeled by an indexl whichvaries from 0 to N−1. An asymptotic analysis using Wiener–Hopf methods indicates that for largeN, thejth component of thelth eigenvector variesroughly in the fashion lnψjl≈iplj+O(1/N). The lth wavevector, pl, varies as for negative values of β and values of l/(N−1) not too close to zero or one. Correspondingly thelth eigenvalue is given by where a is the Fourier transform (also called the symbol) of the Toeplitz matrix.Note that pl has a small positive imaginary part. For values ofj/N not tooclose to zero or one, this imaginary part acts to produce an eigenfunction which decays exponentiallyas j/N increases. Thus, the eigenfunction appears similar to that of a bound state, attached to a wall atj = 0.Near j = 0 this decay is modified by a set of bumps, probably not universal in character. Forj/N above0.6 the eigenfunction begins to oscillate in magnitude and shows deviations from theexponential behavior.The case of 0<α<β<1 need not be studied separately. It can be obtained from the previous one by a ‘conjugacy’ transformationwhich takes ψj into ψN−j−1. This ‘conjugacy’ produces interesting orthonormality relations for the eigenfunctions.