This is a bidiagonal matrix with exponentially decaying entries on the diagonal and 0.5 on the superdiagonal; a small amount of random noise is then added throughout the matrix. This gives it a dense spectrum in a ball around the origin with a few well separated eigenvalues. It was used in [1] as an example of a large matrix for which the pseudospectra of the projected matrix created during the implicitly restarted Arnoldi iteration (ARPACK and eigs) are a good approximation to the pseudospectra of the full matrix. In [1], the dimension N = 200,000 and 30 eigenvalues were requested from a subspace of dimension 50.
[1]: T. G. Wright and L. N. Trefethen, "Computation of pseudospectra using ARPACK and eigs", SIAM J. Sci. Comp., 23(2), 2001, 591-605
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