On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval

TL;DR

This paper derives exact recursive formulas for the probability that all eigenvalues of Gaussian, Wishart, and double Wishart matrices lie within an interval, and shows these probabilities tend to universal constants in large dimensions.

Contribution

It provides efficient recursive formulas for eigenvalue interval probabilities and establishes their asymptotic universal values for large matrices.

Findings

Exact recursive formulas for eigenvalue interval probabilities.

Universal asymptotic probabilities of approximately 0.6921 and 0.9397.

Applications to bounds in compressed sensing.

Abstract

We derive the probability that all eigenvalues of a random matrix $\bf M$ lie within an arbitrary interval $[a,b]$, $\psi(a,b)\triangleq\Pr\{a\leq\lambda_{\min}({\bf M}), \lambda_{\max}({\bf M})\leq b\}$, when $\bf M$ is a real or complex finite dimensional Wishart, double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient recursive formulas allowing the exact evaluation of $\psi(a,b)$ for Wishart matrices, even with large number of variates and degrees of freedom. We also prove that the probability that all eigenvalues are within the limiting spectral support (given by the Mar{\v{c}}enko-Pastur or the semicircle laws) tends for large dimensions to the universal values $0.6921$ and $0.9397$ for the real and complex cases, respectively. Applications include improved bounds for the probability that a Gaussian measurement matrix has a given restricted isometry constant in compressed sensing.