Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a p-variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The

The focus of this paper is on the probability, Eβ(0; J), that a set J consisting of a finite union of intervals contains no eigenvalues for the finite N Gaussian Orthogonal (β = 1) and Gaussian Symplectic (β = 4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (β = 2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function. 1 I.

Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.

We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J J0 is Hilbert-Schmidt, and a proof of Nevai's conjecture that the Szegö condition holds if J J0 is trace class.

This paper surveys the contributions of five mathematicians --- Eugenio Beltrami (1835--1899), Camille Jordan (1838--1921), James Joseph Sylvester (1814--1897), Erhard Schmidt (1876--1959), and Hermann Weyl (1885--1955) --- who were responsible for establishing the existence of the singular value decomposition and developing its theory.

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Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n=p = 1: When centered by p = ( p n 1+ p p) 2 and scaled by p = ( p n 1+ p p)(1= p n 1+1= p p) 1=3 the distribution of x (1) approaches the Tracy-Widom law of order 1, which is dened in terms of the Painleve II dierential equation, and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to ...

The this paper is to describe expansions of solutions to the wave equation on R^n with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in time. To avoid the discussion of particular aspects of potential, gravitational or obstacle scattering, the results are stated using the abstract "black box" formalism of Sjöstrand and the second author [19]. When M is a compact Riemannian manifold, and Delta, its Laplacian, then we have a generalized Fourier expansion of the wave group: sin t p p f(x) = X 2 j 2Spec() e i j t w j (x) ; w j = 2 j w j ; (1.1) and the convergence is absolute in the case of smooth data. The simplest case of a non-compact spectral problem is given by taking R^n with the usual Laplacian outside a compact set -- we can for instance "glue" any compact Riemannian manifold to R^n or consider the obstacle problem with the Dirichlet Laplacian on R^n\O. Since resonances or scattering poles constitute a natural replacement of discrete spectral data for problems on exterior domains, we expect a similar expansion involving them in place of eigenvalues -- this point of view was emphasized early by Lax-Phillips [10] (see [18],[32] and [34] for overviews of recent results). The resonances are de ned as poles of the meromorphic continuation of the resolvent or of the scattering matrix but despite the stationary nature of these de nitions, they are fundamentally a dynamical concept: the real part of a resonance describes the rest energy of a state and the imaginary part its rate of decay. Consequently they should be understood in terms of long time behaviour of solutions to evolutio...

The purpose of this article is to discuss a simple linear algebraic tool which has proved itself very useful in the mathematical study of spectral problems arising in elecromagnetism and quantum mechanics. Roughly speaking it amounts to replacing an operator of interest by a suitably chosen invertible system of operators.

We provide a new proof of a theorem of Birman and Solomyak that if A(s) =A0+ sB with B ≥ 0 trace class and dµs(·) =Tr(B1/2EA(s)(·)B1/2), then ∫ 1 0 [dµs(λ)] ds = ξ(λ) dλ where ξ is the Krein spectral shift from A(0) to A(1). Our main point is that this is a simple consequence of the formula: d ds Tr(f(A(s)) = Tr(Bf ′ (A(s))).