Eigenvalue Distributions in Random Matrices

Introduction

Random matrix theory (RMT) provides a powerful statistical framework for understanding the spectral properties of large, complex systems. When a physical system is characterized by a matrix whose entries are drawn from some probability distribution, the eigenvalue distribution carries deep structural information.

The Semicircle Law

For a Wigner matrix — a symmetric matrix with i.i.d. entries — the empirical eigenvalue distribution converges to the Wigner semicircle law as the matrix dimension grows. This is one of the foundational results in RMT and has direct implications for spectral analysis of signals.

The eigenvalue distribution of a random matrix is not random at all. It converges to a deterministic shape, revealing universal structural properties.

Applications to Signal Analysis

In the TEYA SOL LAB framework, we treat the covariance matrix of a captured signal as a random matrix. The distribution of its eigenvalues tells us:

• The number of significant signal components (information dimension)
• The noise floor of the measurement system
• Structural correlations between measurement channels

Marchenko-Pastur Distribution

When the number of observations and the number of variables grow proportionally, the sample covariance matrix’s eigenvalues follow the Marchenko-Pastur distribution. Deviations from this distribution indicate true signal components that rise above the noise.

The figure shows the theoretical Marchenko-Pastur density (solid line) overlaid on the empirical eigenvalue histogram from a 500×200 random data matrix. The close agreement between theory and empirical data validates the noise model. Eigenvalues beyond the upper edge of the distribution correspond to genuine signal components.

Computational Methods

We implement eigenvalue computation using a combination of numpy.linalg.eigh for exact diagonalization and Lanczos iteration for large sparse systems. The choice depends on the signal dimension and sparsity structure.

import numpy as np

# Generate a Wigner matrix
n = 1000
W = np.random.randn(n, n)
W = (W + W.T) / (2 * np.sqrt(n))

# Compute eigenvalues
eigenvalues = np.linalg.eigvalsh(W)

Conclusions

Random matrix theory provides essential tools for distinguishing signal from noise in spectral measurements. The universality of eigenvalue distributions gives us model-free diagnostics for assessing the information content of physical signals.