Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices
Hui Dai, Zachary Geary, Leo P Kadanoff · Journal of Statistical Mechanics Theory and Experiment · 2009
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. The eigenvalues are labeled by an index l which varies from 0 to N −1. An asymptotic analysis using Wiener–Hopf methods indicates that for large N , the j th component of the l th eigenvector varies roughly in the fashion lnψ j l ≈i p l j +O(1/ N ). The l th wavevector, p l , varies as for negative values of β and values of l /( N −1) not too close to zero or one. Correspondingly the l th eigenvalue is given by where a is the Fourier transform (also called the symbol ) of the Toeplitz matrix. Note that p l has a small positive imaginary part. For values of j / N not too close to zero or one, this imaginary part acts to produce an eigenfunction which decays exponentially as j / N increases. Thus, the eigenfunction appears similar to that of a bound state, attached to a wall at j = 0. Near j = 0 this decay is modified by a set of bumps, probably not universal in character. For j / N above 0.6 the eigenfunction begins to oscillate in magnitude and shows deviations from the exponential behavior. The case of 0<α<β<1 need not be studied separately. It can be obtained from the previous one by a ‘conjugacy’ transformation which takes ψ j into ψ N − j −1 . This ‘conjugacy’ produces interesting orthonormality relations for the eigenfunctions.