The eigenvalues of the pseudospectral second derivative matrix with homogeneous Dirichlet boundary conditions are important in many applications of spectral methods. This paper investigates some of their properties. Numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately, but the errors in the remaining ones are la...

Let X be n×N containing i.i.d. complex entries with E|X11 − EX11| = 1, and T an n× n random Hermitian non-negative definite, independent of X. Assume, almost surely, as n →∞, the empirical distribution function (e.d.f.) of the eigenvalues of T converges in distribution, and the ratio n/N tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of (1/N)XX∗T...

INTRODUCTION: Fractional anisotropy (FA) is a popular scalar measure of voxelwise diffusion characteristics, computed by a nonlinear transformation of the tensor eigenvalues [1]. Despite its ubiquitous use in diffusion MRI studies, the statistical distribution of FA has remained unknown; statistical inference on FA images has either involved a Gaussianity assumption, known to be invalid given t...

A closed analytical expression is derived for the joint distribution function of the real and the imaginary parts of the eigenenergies of the operator H = H0 − iWW+ for the one-channel case, where H0 is taken from the Poissonian or one of the Gaussian ensembles with universality index β, and where the squared moduli |wα |2 of the components of W are assumed to be χ2-distributed with universalit...

Abstract This paper is concerned with the explicit computation of limiting distribution function largest real eigenvalue in Ginibre ensemble when each has been removed independently constant likelihood. We show that recently discovered integrable structures [2] generalize from to its thinned equivalent. Concretely, we express aforementioned as a convex combination two simple Fredholm determinan...

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N+) of a random N x N matrix belonging to Gaussian orthogonal (beta=1), unitary (beta=2) or symplectic (beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N+/N scales, for large N, as P(c,N) approximately = exp[-betaN(2)Phi(c)] where the rate function Ph...