We derive the probability that all eigenvalues of a random matrix M lie within an arbitrary interval [a, b], Ï(a, b) δâ Pra ⤠λmin(M), λmax(M) ⤠b, when M is a real or complex finite-dimensionalWishart, doubleWishart, or Gaussian symmetric/Hermitian matrix. We give efficient recursive formulas allowing the exact evaluation of Ï(a, b) for Wishart matrices, even with a 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 MarÄenko-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.
Chiani, M. (2017). On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval. IEEE TRANSACTIONS ON INFORMATION THEORY, 63(7), 4521-4531 [10.1109/TIT.2017.2694846].
On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval
Chiani, Marco
2017
Abstract
We derive the probability that all eigenvalues of a random matrix M lie within an arbitrary interval [a, b], Ï(a, b) δâ Pra ⤠λmin(M), λmax(M) ⤠b, when M is a real or complex finite-dimensionalWishart, doubleWishart, or Gaussian symmetric/Hermitian matrix. We give efficient recursive formulas allowing the exact evaluation of Ï(a, b) for Wishart matrices, even with a 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 MarÄenko-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.File | Dimensione | Formato | |
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