The Frank matrix is also in the MATLAB gallery, and can be created by the command
F = gallery('frank',N)
This is, a classic example of a matrix whose eigenvalues are ill-conditioned and hence difficult to compute [1,2]. What it illustrates is that one cannot expect to compute eigenvalues better than a set of eps-pseudoeigenvalues for some eps on the order of machine precision. This is seen at the left of the spectrum, where some eigenvalues are complex, even though all the eigenvalues can be proved to be real. The pseudospectra are shown in [3].
[1]: G. H. Golub and J. H. Wilkinson, "Ill-conditioned eigensystems and the computation of the Jordan canonical form", SIAM Review 18, pp. 578-619, 1976.
[2]: J. H. Wilkinson, "The Algebraic Eigenvalue Problem", Clarendon Press, Oxford, 1965.
[3]: L. N. Trefethen, "Psuedospectra of matrices", in "Numerical Analysis 1991" (Dundee 1991), Longman Sci. Tech., Harlow, 1992, 234-266.
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