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

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.