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

We derive the probability $\psi(a,b)=\Pr(a\leq \lambda_{\min}({\bf M}), \lambda_{\max}({\bf M})\leq b)$ that all eigenvalues of a random matrix $\bf M$ lie within an arbitrary interval $[a,b]$, when $\bf M$ is a real or complex finite dimensional Wishart, double Wishart, or Gaussian symmetric/hermitian matrix. We give efficient recursive formulas allowing, for instance, the exact evaluation of $\psi(a,b)$ for Wishart matrices with number of variates $500$ and degrees of freedom $1000$. We also prove that the probability that all eigenvalues are within the limiting spectral support (given by the Marchenko-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.