This is a random matrix whose entries are drawn from the normal distribution with mean 0 and variance 1/N.
One can see from these pseudospectra that dense random matrices are only mildly non-normal (in contrast, for example, to triangular ones). In the limit N -> Inf, the norm and spectral abscissa converge to 2 and 1, respectively [1,2] and the condition number is of size O(N) [3]. Of that factor of N, O(sqrt(N)) can be attributed to variation in the size of the eigenvalues and O(sqrt(N)) to non-normality [4]. Pseudospectra for this example were first plotted in [5].
[1]: S. Geman, "A limit theorem for the norm of random matrices" The Annals of Probability 8(2), 1980, 252-261.
[2]: S. Geman, "The spectral radius of large random matrices" The Annals of Probability 14(4), 1986, 1318-1328.
[3]: A. Edelman, "Eigenvalues and Condition Numbers of Random Matrices", SIAM J. Matrix Anal. 9(4), 1988, 543-560.
[4]: J. T. Chalker and B. Mehlig, "Eigenvector Statistics in Non- Hermitian Random Matrix Ensembles", Phys. Rev. Lett. 81(16), 1998, 3367-3370.
[5]: L. N. Trefethen, "Psuedospectra of matrices", in "Numerical Analysis 1991" (Dundee 1991), Longman Sci. Tech., Harlow, 1992, 234-266.
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