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# Complex Matrix

A matrix whose elements may contain complex numbers.

The matrix product of two complex matrices is given by

 (1)

where

 (2) (3) (4) (5) (6) (7) (8) (9)

Hadamard (1893) proved that the determinant of any complex matrix with entries in the closed unit disk satisfies

 (10)

(Hadamard's maximum determinant problem), with equality attained by the Vandermonde matrix of the roots of unity (Faddeev and Sominskii 1965, p. 331; Brenner 1972). The first few values for , 2, ... are 1, 2, , 16, , 216, ....

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 for , , 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).

Complex Vector, Hadamard's Maximum Determinant Problem, Integer Matrix, k-Matrix, Matrix, Real Matrix

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## References

Brenner, J. and Cummings, L. "The Hadamard Maximum Determinant Problem." Amer. Math. Monthly 79, 626-630, 1972.Edelman, A. "The Probability that a Random Real Gaussian Matrix has Real Eigenvalues, Related Distributions, and the Circular Law." J. Multivariate Anal. 60, 203-232, 1997.Faddeev, 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." In Random Matrices and Their Applications. Providence, RI: Amer. Math. Soc., pp. 145-152, 1986.Kaltofen, E. "Challenges of Symbolic Computation: My Favorite Open Problems." J. Symb. Comput. 29, 891-919, 2000.Mehta, M. L. Random Matrices, 3rd ed. New York: Academic Press, 2004.Poljak, S. and Rohn, J. "Checking Robust Nonsingularity is NP-Hard." Math. Control Signals Systems 6, 1-9, 1993.

Complex Matrix

## Cite this as:

Weisstein, Eric W. "Complex Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ComplexMatrix.html