Random Matrix

A random matrix is a matrix of given type and size whose entries are random variables with a specified joint statistical distribution.

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

For a real matrix of size , where is a positive integer, with independent entries having a standard normal distribution, the expected number of real eigenvalues is given by

			(1)
			(2)

where is a hypergeometric function and is a beta function (Edelman et al. 1994, Edelman and Kostlan 1995). has asymptotic behavior

(3)

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

(4)

where unless and is even. For , the full probability function for the number of real eigenvalues was derived by Kanzieper and Akemann (2005) as

(5)

where

			(6)
			(7)

The sum runs over all partitions of size , where is the number of pairs of complex-conjugate eigenvalues, and is a zonal polynomial. The frequency representation has distinct parts , with respective multiplicities , and (Kanzieper and Akemann 2005). The arguments depend on the parity of (the matrix dimension) and are given by

(8)

where is a positive integer, is a matrix trace, and is an matrix with entries

						(9)
						(10)

where and vary between 0 and and , with the floor function. The extra column used for odd is evaluated by the same integral formula. Here, are generalized Laguerre polynomials, with , and is the erfc, the complementary error function (Kanzieper and Akemann 2005).

For , Edelman (1997) proved that the density of nonreal eigenvalues in the upper half-plane of such a matrix is

			(11)
			(12)

for , where is the erfc (complementary error) function, is the exponential sum function, and is the upper incomplete gamma function. The integral of over a region is the expected number of eigenvalues there. Integrating over the upper half-plane (and multiplying by 2) gives the expected number of nonreal eigenvalues as

			(13)
			(14)
			(15)

where is a regularized hypergeometric function (Edelman 1997). The first few values are

			(16)
			(17)
			(18)
			(19)
			(20)

(OEIS A052928, A093605, and A046161).

Girko's circular law implies that if is chosen uniformly from the eigenvalues of such a matrix, then converges in distribution to the uniform distribution on the unit disk in the complex plane as (Edelman 1997, p. 209).

Wigner's semicircle law states that the limiting statistical distribution of the eigenvalues of a Wigner matrix with off-diagonal entry variance 1, scaled by , is the semicircular distribution (Tao 2010).

If matrices are chosen independently with probability 1/2 from each of

			(21)
			(22)

then, with probability one,

(23)

where (OEIS A078416) and denotes the matrix spectral norm (Bougerol and Lacroix 1985, pp. 11 and 157; Viswanath 2000). The value is the constant appearing in the random Fibonacci sequence. The following Wolfram Language code can be used to estimate , rescaling after each multiplication to avoid numerical overflow.

With[{n = 100000},
 Exp[
 Last[
 Fold[
 Function[{state, sign},
 Module[{m, scale},
 m = state[[1]] . {{0., 1.}, {1., sign}};
 scale = Norm[m, 2];
 {m/scale, state[[2]] + Log[scale]}
 ]
 ],
 {N[IdentityMatrix[2]], 0.},
 RandomChoice[{-1, 1}, {n}]
 ]
 ]/n
 ]
 ]

For random real symmetric matrices, Liang et al. (2026) proved a uniform matrix rank bound under a bound on point probabilities. Let be an random real symmetric matrix whose entries above the diagonal are independent and have every point probability at most a fixed . The diagonal entries may be jointly dependent, but their joint vector is independent of the off-diagonal entries. There is a constant such that

(24)

for . No moment, centering, or identical-distribution assumption is imposed. Liang et al. (2026) report AI assistance in checking and refining technical details of the proof.