This matrix is the Chebyshev spectral differentiation matrix of dimension N [1] of the first derivative operator d/dx on [-1,1] with boundary condition u(1)=0. The pseudospectra of the infinite dimensional operator consist of half-planes, which is suggested in this low-dimensional discretization by the straight lines that form the rightmost part of each pseudospectral boundary [2].
[1]: L. N. Trefethen, "Spectral methods in MATLAB", SIAM, 2000.
[2]: L. N. Trefethen, "Psuedospectra of matrices", in "Numerical Analysis 1991" (Dundee 1991), Longman Sci. Tech., Harlow, 1992, 234-266.
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