Over a million developers have joined DZone.
{{announcement.body}}
{{announcement.title}}

Girko-Ginibri Circular Law for Random Matrix Eigenvalues

DZone's Guide to

Girko-Ginibri Circular Law for Random Matrix Eigenvalues

A math whiz and programming expert looks at how to work with random data sets using the Girko-Ginibri Circular Law, and implements it in Python.

· Big Data Zone ·
Free Resource

The open source HPCC Systems platform is a proven, easy to use solution for managing data at scale. Visit our Easy Guide to learn more about this completely free platform, test drive some code in the online Playground, and get started today.

Suppose you create a large matrix by filling its components with random values. If M has size n by n, then we require the probability distribution for each entry to have mean 0 and variance 1/n. Then the Girko-Ginibri circular law says that the eigenvalues of M are approximately uniformly distributed in the unit disk in the complex plane. As the size n increases, the distribution converges to a uniform distribution on the unit disk.

The probability distribution need not be normal. It can be any distribution, shifted to have mean 0 and scaled to have variance 1/n, provided the tail of the distribution isn't so thick that the variance doesn't exist. If you don't scale the variance to 1/n you still get a circle, just not a unit circle.

We'll illustrate the circular law with a uniform distribution. The uniform distribution has mean 1/2 and variance 1/12, so we will subtract 1/2 and multiply each entry by √(12/n).

Here's our Python code:

    import matplotlib.pyplot as plt
    import numpy as np

    n = 100
    M = np.random.random((n,n)) - 0.5
    M *= (12/n)**0.5
    w = np.linalg.eigvals(M)
    plt.scatter(np.real(w), np.imag(w))
    plt.axes().set_aspect(1)
    plt.show()

When n=100 we get the following plot.

When n=1000 we can see the disk filling in more.

Note that the points are symmetric about the real axis. All the entries of M are real, so its characteristic polynomial has all real coefficients, and so its roots come in conjugate pairs. If we randomly generated complex entries for M we would not have such symmetry.

Managing data at scale doesn’t have to be hard. Find out how the completely free, open source HPCC Systems platform makes it easier to update, easier to program, easier to integrate data, and easier to manage clusters. Download and get started today.

Topics:
python ,big data ,random data ,data visualization ,tutorial

Published at DZone with permission of

Opinions expressed by DZone contributors are their own.

{{ parent.title || parent.header.title}}

{{ parent.tldr }}

{{ parent.urlSource.name }}