This is a Markov chain transition matrix for a random walk on an N by N triangular lattice. It is an example of a matrix for which certain parts of the spectrum (in this case the interior) are highly non-normal, but other eigenvalues are well-conditioned and relevant to the application. (Here, the left eigenvectors associated with eigenvalues of unit modulus determine the steady-state distribution of the Markov chain. For more information, see [1].
[1]: Y. Saad, "Numerical methods for large eigenvalue problems", Manchester University Press, Manchester, p48ff, 1992.
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