Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices (original) (raw)

The Eigenvectors of Single Spiked Complex Wishart Matrices Finite and Asymptotic Analyses 1 1

Let W ∈ C n×n be a single-spiked Wishart matrix in the class W ∼ CWn (m, In + θvv †) with m ≥ n, where In is the n × n identity matrix, v ∈ C n×1 is an arbitrary vector with unit Euclidean norm, θ ≥ 0 is a non-random parameter, and (•) † represents the conjugate-transpose operator. Let u1 and un denote the eigenvectors corresponding to the smallest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity Z (n) = ¬ ¬ v † u ¬ ¬ 2 ∈ (0, 1) for = 1, n. In particular, we derive a finite dimensional closed-form p.d.f. for Z (n) 1 which is amenable to asymptotic analysis as m, n diverges with m − n fixed. It turns out that, in this asymptotic regime, the scaled random variable nZ (n) 1 converges in distribution to χ 2 2 /2(1 + θ), where χ 2 2 denotes a chi-squared random variable with two degrees of freedom. This reveals that u1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of Z (n) n is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension (n − 2). Although a simple solution to this double integral seems intractable, for special configurations of n = 2, 3, and 4, we obtain closed-form expressions.

Large Complex Correlated Wishart Matrices: Fluctuations and Asymptotic Independence at the Edges

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely towards that edge and fluctuates according to the Tracy-Widom law at the scale N2/3N^{2/3}N2/3. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin, we prove that the smallest eigenvalue fluctuates according to the hard-edge Tracy-Widom law at the scale N2N^2N2. As an application, an asymptotic study of the condition number of large correlated Wishart matrices is provided.

The Outliers Among The Singular Values Of Large Rectangular Random Matrices With Additive Fixed Rank Deformation

2014

Consider the matrix Σn=n−1/2XnD1/2n+Pn where the matrix XninCNtimesnX_n \in \C^{N\times n}XninCNtimesn has Gaussian standard independent elements, Dn is a deterministic diagonal nonnegative matrix, and Pn is a deterministic matrix with fixed rank. Under some known conditions, the spectral measures of ΣnΣ∗n and n−1XnDnX∗n both converge towards a compactly supported probability measure μ as N,n→∞ with N/n→c>0. In this paper, it is proved that fini...

Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III Double scaling analysis DATASET in JOURNAL OF MATHEMATICAL PHYSICS · JUNE 2015 5 PUBLICATIONS 3 CITATIONS

We continue with the study of the Hankel determinant, defined by D n (t, α) = det  ∞ 0 x j+k w(x; t, α)dx n−1 j,k=0 , generated by a singularly perturbed Laguerre weight, w(x; t, α) = x α e −x e −t /x , x ∈ R + , α > 0, t > 0, and obtained through a deformation of the Laguerre weight function, w(x; 0, α) = x α e −x , x ∈ R + , α > 0, via the multiplicative factor e −t /x . An earlier investigation was made on the finite n aspect of such determinants, which appeared in Chen and Its [J. Approx. Theory 162, 270-297 (2010)]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painlevé III (P III , for short) transcendent and its t derivatives. In this paper, we show that under a double scaling, where n, the size of the Hankel matrix tends to ∞, and t tends to 0 + , the scaled-and therefore, in some sense, infinite dimensional-Hankel determinant has an integral representation in terms of a C potential. The second order non-linear ordinary differential equation satisfied by C, after a change of variables, is another P III transcendent, albeit with fewer number of parameters. Expansions of the double scaled determinant for small and large parameters are obtained. C 2015 AIP Publishing LLC.

Extreme eigenvalue distributions of Jacobi ensembles: New exact representations, asymptotics and finite size corrections

Nuclear Physics B, 2019

Let W 1 and W 2 be independent n×n complex central Wishart matrices with m 1 and m 2 degrees of freedom respectively. This paper is concerned with the extreme eigenvalue distributions of double-Wishart matrices (W 1 + W 2) −1 W 1 , which are analogous to those of F matrices W 1 W −1 2 and those of the Jacobi unitary ensemble (JUE). Defining α 1 = m 1 − n and α 2 = m 2 − n, we derive new exact distribution formulas in terms of (α 1 + α 2)-dimensional matrix determinants, with elements involving derivatives of Legendre polynomials. This provides a convenient exact representation, while facilitating a direct large-n analysis with α 1 and α 2 fixed (i.e., under the so-called "hard-edge" scaling limit); the analysis is based on new asymptotic properties of Legendre polynomials and their relation with Bessel functions that are here established. Specifically, we present limiting formulas for the smallest and largest eigenvalue distributions as n → ∞ in terms of α 1-and α 2-dimensional determinants respectively, which agrees with expectations from known universality results involving the JUE and the Laguerre unitary ensemble (LUE). We also derive finite-n corrections for the asymptotic extreme eigenvalue distributions under hard-edge scaling, giving new insights on universality by comparing with corresponding correction terms derived recently for the LUE. Our derivations are based on elementary algebraic manipulations, differing from existing results on double-Wishart and related models which often involve Fredholm determinants, Painlevé differential equations, or hypergeometric functions of matrix arguments.

Characterizations of the distribution of the Demmel condition number of real Wishart matrices

Special Matrices, 2016

The Demmel condition number is an indicator of the matrix condition, and its properties have recently found applications in many practical problems, such as in MIMO communication systems, in the analytical prediction of level-crossing and fade duration statistics of Rayleigh channels, and in spectrum sensing for cognitive radio systems, among others. As the characterizations of a probability distribution play an important role in probability and statistics, in this paper we study the characterizations of the distribution of the Demmel condition number of real Wishart matrices by truncated first moment. Since the truncated distributions arise in practical statistics where the ability of record observations is limited to a given threshold or within a specified range, we hope that these characterizations will be quite useful for practitioners and researchers in the fields of probability, statistics, and other applied sciences, such as actuarial science, linear algebra, multivariate sta...

Distributions of Demmel and Related Condition Numbers

2012

Consider a random matrix A ∈ C m×n (m ≥ n) containing independent complex Gaussian entries with zero mean and unit variance, and let 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λn < ∞ denote the eigenvalues of A * A, where (·) * represents conjugate-transpose. This paper investigates the distribution of the random variables n j=1 λ j λ k for k = 1 and k = 2. These two variables are related to certain condition number metrics, including the so-called Demmel condition number, which have been shown to arise in a variety of applications. For both cases, we derive new exact expressions for the probability densities and establish the asymptotic behavior as the matrix dimensions grow large. In particular, it is shown that as n and m tend to infinity with their difference fixed, both densities scale on the order of n 3 . After suitable transformations, we establish exact expressions for the asymptotic densities, obtaining simple closed-form expressions in some cases. Our results generalize the work of Edelman on the Demmel condition number for the case m = n.

Singular linear statistics of the Laguerre unitary ensemble and Painlevé. III. Double scaling analysis

Journal of Mathematical Physics, 2015

We continue with the study of the Hankel determinant, defined by, D n (t, α) = det ∞ 0 x j+k w(x; t, α)dx n−1 j,k=0 , generated by a singularly perturbed Laguerre weight, w(x; t, α) = x α e −x e −t/x , x ∈ R + , α > 0, t > 0, obtained through a deformation of the Laguerre weight function, w(x; 0, α) = x α e −x , x ∈ R + , α > 0, via the multiplicative factor e −t/x. An earlier investigation was made on the finite n aspect of such determinants, which appeared in [20]. It was found that the logarithm of the Hankel determinant has an integral representation in terms of a particular Painlevé III(P III , for short) and its t derivatives. In this paper we show that, under a double scaling, where n , the order of the Hankel matrix tends to ∞, and t , tends to 0 + , the scaled-and therefore, in some sense, infinite dimensional-Hankel determinant, has an integral representation in terms of a C potential. The second order non-linear ode satisfied by C, after a change of variable, is another P III transcendent, albeit with fewer number of parameters. Expansions of the double scaled determinant for small and large parameter are obtained.

Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices (original) (raw)

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