in this paper, the ultraspherical operational matrices of derivatives are constructed. based on these operational matrices, two numerical algorithms are presented and analyzed for obtaining new approximate spectral solutions of a class of linear and nonlinear lane-emden type singular initial value problems. the basic idea behind the suggested algorithms is basically built on transforming the eq...

Here, we will consider one approach for extending the ideas underlying the least-squares algorithm we discussed in class to non-tall matrices. Let A ∈ Rn×d matrix, where both n and d are large, and where rank(A) = k exactly, and let B ∈ Rn×t. Consider the problem minX∈Rn×t‖AX −B‖, where ‖ · ‖ is a unitarily-invariant matrix norm. The solution to this problem is Xopt = AB, and here we consider a...

We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider n× n matrices with complex entries of the form M = A ◦X +B = (aijξij + bij) where X = (ξij) has iid centered entries of unit variance and A and B are fixed matrices. In our main result we obtain polynomial bounds on the smallest si...

Nir Avni Counting representations of arithmetic groups I’ll talk about the rate of growth of the number of d-dimensional representations of a higher-rank arithmetic group G(Z), as d tends to infinity. This asymptotics is related to the distribution of random commutators in the congruence quotients of G(Z), the singular values of random matrices in Lie(G), and the geometry of the character varie...

In statistics and machine learning, people are often interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. One usually employs Davis-Kahan sin θ theorem to bound the difference between the eigenvecto...

Abstract—This paper investigates the behaviour of the spectrum of generally correlated Gaussian random matrices whose columns are zero-mean independent vectors but have different correlations, under the specific regime where the number of their columns and that of their rows grow at infinity with the same pace. This work is, in particular, motivated by applications from statistical signal proce...

The paper consists of two parts. In the first part, new canonical forms are defined for singular 1D linear systems and a procedure to determine nonsingular matrices transforming matrices of singular systems to their canonical forms is derived. In the second part new canonical forms of matrices of the singular 2D Roesser model are defined and a procedure for determining realisations in canonical...

In this paper, the ultraspherical operational matrices of derivatives are constructed. Based on these operational matrices, two numerical algorithms are presented and analyzed for obtaining new approximate spectral solutions of a class of linear and nonlinear Lane-Emden type singular initial value problems. The basic idea behind the suggested algorithms is basically built on transforming the eq...

In this paper, we consider some inverse singular value problems for Toeplitz-related matrices. We construct a Toeplitz-plus-Hankel matrix from prescribed singular values including a zero singular value. Then we find a solution to the inverse singular value problem for Toeplitz matrices which have double singular values including a double zero singular value.

We investigate random, discrete Schrödinger operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson’s Coulomb gas inverse temperature β. They are similar to the class of “critical” random Schrödiner operators with random potentials which diminish as |x|− 12 . We show that as a function of β they undergo a transition from a regime of (power-law) local...