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Fundamental Laws of Random Matrix Theory

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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.

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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 .

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Contributed by: Jessica Alfonsi  (January 2023)
(Padova, Italy)
Open content licensed under CC BY-NC-SA

Snapshots

Details

Snapshot 1: plot corresponding to the circular law and diagonalization of a random matrix

Snapshot 2: histogram density plot corresponding to Wigner semicircle law obtained by diagonalization of a symmetrized random matrix

Snapshot 3: histogram density plot corresponding to Marcenko–Pastur law obtained by diagonalization of the scalar product of a rectangular matrix

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