The "cross" matrix is an example of a banded "twisted circulant matrix". It is tridiagonal (with periodic wraparound) but instead of having constant diagonals, as in a true circulant matrix, it has a smoothly varying main diagonal. The result is an exponentially strong degree of nonnormality, with pseudomodes in the form of wave packets. See [1].
[1]: J. Chapman and L. N. Trefethen, "Wave packet pseudomodes of twisted Toeplitz matrices", to appear.
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