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

Abstract

We derive the probability that all eigenvalues of a random matrix $\mathbf{M}$ lie within an arbitrary interval $[a,b]$, $\psi (a,b)\triangleq \mathrm{Pr}\{a\le {\lambda }_{\min }(\mathbf{M}),{\lambda }_{\max }(\mathbf{M})\le b\}$, when $\mathbf{M}$ 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{č}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.

IEEE Transactions on Information Theory, 2017

References in corpus (11)

• Large Deviations of Extreme Eigenvalues of Random Matrices
• Finite sample approximation results for principal component analysis: a matrix perturbation approach
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Cooperative Spectrum Sensing based on the Limiting Eigenvalue Ratio Distribution in Wishart Matrices
• Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
• Cosmology From Random Multifield Potentials
• Accuracy of the Tracy--Widom limits for the extreme eigenvalues in white Wishart matrices
• A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix
• Charting an Inflationary Landscape with Random Matrix Theory
• Approximate null distribution of the largest root in multivariate analysis
• Distribution of the largest root of a matrix for Roy's test in multivariate analysis of variance

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