Fundamental Laws of Random Matrix Theory

RMT law

circular

Wigner semicircle

Marcenko-Pastur

matrix size

250

500

750

1000

size ratio for matrix B

This Demonstration exhibits the three fundamental laws of random matrix theory related to the eigenvalue distributions for a selected matrix transformation. Starting from the eigenvalues of an

n×n

random matrix

with its elements distributed according to the normal distribution with zero mean and unit variance, we can verify convergence to the circular law in the limit as

n∞

, appropriately rescaled by a factor

. Thus the limiting spectral distribution is the uniform distribution over the unit disk in the complex plane (see Related Links). You can see this by selecting the "circular" button, for

eig(A)

. You can set the matrix size by selecting the corresponding button in the second row.

By computing the eigenvalues of the corresponding symmetrized random matrix

(A+

†

)

and rescaling them by the factor

, we can verify that the histogram density distribution follows the Wigner semicircle law (dashed blue curve). Click the "Wigner semicircle" button for

eig(A+

†

)

(see Related Links for further applications).

To verify the Marcenko–Pastur law, we start from a random rectangular matrix

of size

n×m

where

can divide

. The control "size ratio for rectangular matrix

" is enabled only when the "Marcenko–Pastur" button for

eig(

†

) is selected, which lets you adjust the

n/m

ratio. By computing the scaled-by-

eigenvalues of the matrix product

†

and plotting their histogram density distribution, we can verify that this follows the Marcenko–Pastur distribution (highlighted in red) with dispersion

σ=1

and

λ=1/m