The Grcar matrix is also in the MATLAB gallery, and can be created by the command
G = gallery('grcar',N)
This is a popular example in the field of matrix iterations of a matrix whose spectrum is in the right half-plane but whose numerical range is not. It's also a popular example in the study of nonsymmetric Toeplitz matrices. The matrix was first described in [1] and its pseudospectra were first plotted in [2].
[1]: J. F. Grcar, "Operator coefficient methods for linear equations", tech. report SAND89-8691, Sandia National Labs, 1989
[2]: L. N. Trefethen, "Psuedospectra of matrices", in "Numerical Analysis 1991" (Dundee 1991), Longman Sci. Tech., Harlow, 1992, 234-266.
EigTool EigTool Demos Menu.