Random Matrix

A random matrix is a matrix of given type and size whose entries consist of random numbers from some specified distribution.

Random matrix theory is cited as one of the "modern tools" used in Catherine's proof of an important result in prime number theory in the 2005 film Proof.

For a real n×n matrix with elements having a standard normal distribution, the expected number of real eigenvalues is given by

={sqrt(2)sum_(k=0)^(n/2-1)((4k-1)!!)/((4k)!!) for n even; 1+sqrt(2)sum_(k=1)^((n-1)/2)((4k-3)!!)/((4k-2)!!) for n odd,

where _2F_1(a,b;c;z) is a hypergeometric function and B(z,a) is a beta function (Edelman et al. 1994, Edelman and Kostlan 1994). E_n has asymptotic behavior


Let p_(n,k) be the probability that there are exactly k real eigenvalues in the complex spectrum of the n×n matrix. Edelman (1997) showed that


which is the smallest probability of all p_(n,k)s. The entire probability function of the number of expected real eigenvalues in the spectrum of a Gaussian real random matrix was derived by Kanzieper and Akemann (2005) as




In (6), the summation runs over all partitions lambda of length l, l is the number of pairs of complex-conjugated eigenvalues, and Z_((1^l)) are zonal polynomial. In addition, (6) makes use a frequency representation of the partition lambda (Kanzieper and Akemann 2005). The arguments p_l depend on the parity of n (the matrix dimension) and are given by


where Tr(A) is a matrix trace, rho^^ is an m×m matrix with entries


alpha and beta vary between 0 and m-1, m=|_n/2_| with |_x_| the floor function), L_j^alpha(z) are generalized Laguerre polynomials, and erfc(z) is the complementary erf function erfc (Kanzieper and Akemann 2005).


Edelman (1997) proved that the density of a random complex pair of eigenvalues x+/-iy of a real n×n matrix whose elements are taken from a standard normal distribution is


for y>=0, where erfc(z) is the erfc (complementary error) function, e_n(z) is the exponential sum function, and Gamma(a,x) is the upper incomplete gamma function. Integrating over the upper half-plane (and multiplying by 2) gives the expected number of complex eigenvalues as


(Edelman 1997). The first few values are


(OEIS A052928, A093605, and A046161).

Girko's circular law considers eigenvalues lambda (possibly complex) of a set of random n×n real matrices with entries independent and taken from a standard normal distribution and states that as n->infty, lambda/sqrt(n) is uniformly distributed on the unit disk in the complex plane.

Wigner's semicircle law states that the for large n×n symmetric real matrices with elements taken from a distribution satisfying certain rather general properties, the distribution of eigenvalues is the semicircle function.

If n matrices M_i are chosen with probability 1/2 from one of

M_+=[0 1; 1 1]
M_-=[0 1; 1 -1],



where e^c=1.13198824... (OEIS A078416) and ||M|| denotes the matrix spectral norm (Bougerol and Lacroix 1985, pp. 11 and 157; Viswanath 2000). This is the same constant appearing in the random Fibonacci sequence. The following Wolfram Language code can be used to estimate this constant.

  With[{n = 100000},
    m = Fold[Dot, IdentityMatrix[2],
      {{0, 1}, {1, #}}& /@
        RandomChoice[{-1, 1}, {n}]
    ] // N;
    Log[Sqrt[Max[Eigenvalues[Transpose[m] . m]]]] /

See also

Complex Matrix, Girko's Circular Law, Integer Matrix, Matrix, Random Fibonacci Sequence, Real Matrix, Wigner's Semicircle Law

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Bougerol, P. and Lacroix, J. Random Products of Matrices with Applications to Schrödinger Operators. Basel, Switzerland: Birkhäuser 1985.Chassaing, P.; Letac, G.; and Mora, M. "Brocot Sequences and Random Walks on SL_2(R)." In Probability Measures on Groups VII (Ed. H. Heyer). New York Springer-Verlag, pp. 36-48, 1984.Edelman, A. "The Probability that a Random Real Gaussian Matrix has k Real Eigenvalues, Related Distributions, and the Circular Law." J. Multivariate Anal. 60, 203-232, 1997.Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial are Real?" Bull. Amer. Math. Soc. 32, 1-37, 1995.Edelman, A.; Kostlan, E.; and Shub, M. "How Many Eigenvalues of a Random Matrix are Real?" J. Amer. Math. Soc. 7, 247-267, 1994.Furstenberg, H. "Non-Commuting Random Products." Trans. Amer. Math. Soc. 108, 377-428, 1963.Furstenberg, H. and Kesten, H. "Products of Random Matrices." Ann. Math. Stat. 31, 457-469, 1960.Girko, V. L. Theory of Random Determinants. Boston, MA: Kluwer, 1990.Kanzieper, E. and Akemann, G. "Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices." Phys. Rev. Lett. 95, 230201-1-230201-4, 2005.Katz, M. and Sarnak, P. Random Matrices, Frobenius Eigenvalues, and Monodromy. Providence, RI: Amer. Math. Soc., 1999.Lehmann, N. and Sommers, H.-J. "Eigenvalue Statistics of Random Real Matrices." Phys. Rev. Let. 67, 941-944, 1991.Mehta, M. L. Random Matrices, 3rd ed. New York: Academic Press, 1991.Sloane, N. J. A. Sequences A046161, A052928, A078416, and A093605 in "The On-Line Encyclopedia of Integer Sequences."Viswanath, D. "Random Fibonacci Sequences and the Number 1.13198824...." Math. Comput. 69, 1131-1155, 2000.

Referenced on Wolfram|Alpha

Random Matrix

Cite this as:

Weisstein, Eric W. "Random Matrix." From MathWorld--A Wolfram Web Resource.

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