Studying the maximum possible eigenvalue norms for random complex matrices is computationally intractable. Although average
properties of the distribution of can be determined, finding the maximum value corresponds
to determining if the set of matrices contains a singular
matrix, which has been proven to be an NP-complete
problem (Poljak and Rohn 1993, Kaltofen 2000). The above plots show the distributions
, and matrix eigenvalue norms for elements uniformly distributed
inside the unit disk .
Similar plots are obtained for elements uniformly distributed inside . The exact distribution of eigenvalues for
complex matrices with both real and imaginary parts distributed as independent standard
normal variates is given by Ginibre (1965), Hwang (1986), and Mehta (1991).
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D. K. and Sominskii, I. S. Problems
in Higher Algebra. San Francisco: W. H. Freeman, 1965.Ginibre,
J. "Statistical Ensembles of Complex, Quaternion, and Real Matrices." J.
Math. Phys.6, 440-449, 1965.Hadamard, J. "Résolution
d'une question relative aux déterminants." Bull. Sci. Math.17,
30-31, 1893.Hwang, C. R. "A Brief Survey on the Spectral Radius
and the Spectral Distribution of Large Random Matrices with i.i.d. Entries."
Matrices and Their Applications. Providence, RI: Amer. Math. Soc., pp. 145-152,
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Open Problems." J. Symb. Comput.29, 891-919, 2000.Mehta,
M. L. Random
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