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

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

, 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

. 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

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

(see Related Links for further applications).

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

of size

where

can divide

. The control "size ratio for rectangular matrix

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

) is selected, which lets you adjust the

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

and

.