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.Comment: IEEE Transactions on Information Theory, 201

Eigenvalue Dynamics of a Central Wishart Matrix with Application to MIMO Systems

We investigate the dynamic behavior of the stationary random process defined by a central complex Wishart (CW) matrix ${\bf{W}}(t)$ as it varies along a certain dimension $t$. We characterize the second-order joint cdf of the largest eigenvalue, and the second-order joint cdf of the smallest eigenvalue of this matrix. We show that both cdfs can be expressed in exact closed-form in terms of a finite number of well-known special functions in the context of communication theory. As a direct application, we investigate the dynamic behavior of the parallel channels associated with multiple-input multiple-output (MIMO) systems in the presence of Rayleigh fading. Studying the complex random matrix that defines the MIMO channel, we characterize the second-order joint cdf of the signal-to-noise ratio (SNR) for the best and worst channels. We use these results to study the rate of change of MIMO parallel channels, using different performance metrics. For a given value of the MIMO channel correlation coefficient, we observe how the SNR associated with the best parallel channel changes slower than the SNR of the worst channel. This different dynamic behavior is much more appreciable when the number of transmit ($N_T$) and receive ($N_R$) antennas is similar. However, as $N_T$ is increased while keeping $N_R$ fixed, we see how the best and worst channels tend to have a similar rate of change.Comment: 15 pages, 9 figures and 1 table. This work has been accepted for publication at IEEE Trans. Inf. Theory. Copyright (c) 2014 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]

Extreme Eigenvalue Distributions of Some Complex Correlated Non-Central Wishart and Gamma-Wishart Random Matrices

Let $\mathbf{W}$ be a correlated complex non-central Wishart matrix defined through $\mathbf{W}=\mathbf{X}^H\mathbf{X}$, where $\mathbf{X}$ is $n\times m \, (n\geq m)$ complex Gaussian with non-zero mean $\boldsymbol{\Upsilon}$ and non-trivial covariance $\boldsymbol{\Sigma}$. We derive exact expressions for the cumulative distribution functions (c.d.f.s) of the extreme eigenvalues (i.e., maximum and minimum) of $\mathbf{W}$ for some particular cases. These results are quite simple, involving rapidly converging infinite series, and apply for the practically important case where $\boldsymbol{\Upsilon}$ has rank one. We also derive analogous results for a certain class of gamma-Wishart random matrices, for which $\boldsymbol{\Upsilon}^H\boldsymbol{\Upsilon}$ follows a matrix-variate gamma distribution. The eigenvalue distributions in this paper have various applications to wireless communication systems, and arise in other fields such as econometrics, statistical physics, and multivariate statistics.Comment: Accepted for publication in Journal of Multivariate Analysi

Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices

One of the key issues in the acquisition of sparse data by means of compressed sensing (CS) is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the required number of measurements for sparse recovery. In this paper we provide a new approach for the analysis of the restricted isometry constant (RIC) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distributions of the extreme eigenvalues for Wishart matrices. First, we derive the probability that the restricted isometry property is satisfied for a given sufficient recovery condition on the RIC, and propose a probabilistic framework to study both the symmetric and asymmetric RICs. Then, we analyze the recovery of compressible signals in noise through the statistical characterization of stability and robustness. The presented framework determines limits on various sparse recovery algorithms for finite size problems. In particular, it provides a tight lower bound on the maximum sparsity order of the acquired data allowing signal recovery with a given target probability. Also, we derive simple approximations for the RICs based on the Tracy-Widom distribution.Comment: 11 pages, 6 figures, accepted for publication in IEEE transactions on information theor

Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt eigenvalues. For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint. We first compute the distribution and moments of the smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.Comment: 25 pag., 4 fig - minor changes, typos fixed. To appear in JSTA

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N {\em strongly correlated} random variables for all values of N (and not just for large N).Comment: 13 pages, 2 figures included; typos corrected; to appear in J. Stat. Phy