Related papers: How well-conditioned can the eigenvalue problem be…
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
We estimate the condition number of a matrix by applying the Power Method, that is essentially a sequence of matrix-by-vector multiplications, similarly to the Lanczos method.
Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus,…
In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix…
In the first part of these notes, we review some of the recent developments in the study of the spectral properties of Wigner matrices. In the second part, we present a new proof of a Wegner estimate for the eigenvalues of a large class of…
We consider optimization problems with manifold-valued constraints. These generalize classical equality and inequality constraints to a setting in which both the domain and the codomain of the constraint mapping are smooth manifolds. We…
We consider the first eigenvalues of the polyharmonic, Lam\'e and Stokes operators with Dirichlet boundary conditions on sets of given finite measure. It is shown that a quasi-open set for which this eigenvalue is minimal is open. This…
In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First,…
It is shown that the eigenvalues $\lambda_k, k=1, 2, \dots,$ of the one-particle density matrix satisfy the bound $\lambda_k\le C k^{-8/3}$ with a positive constant $C$.
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative…
Let $n$ be a positive integer and $m$ be a positive even integer. Let ${\mathcal A}$ be an $m^{th}$ order $n$-dimensional real weakly symmetric tensor and ${\mathcal B}$ be a real weakly symmetric positive definite tensor of the same size.…
The formulation of the eigenvalue problem for the Schr\"odinger equation is studied, for the numerical solution a new approach is applied. With the usual exponentially rising free-state asymptotical behavior, and also with a first order…
Matrix functions play an increasingly important role in many areas of scientific computing and engineering disciplines. In such real-world applications, algorithms working in floating-point arithmetic are used for computing matrix functions…
It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in $\R^p$, where $p$ is the…
Various notions of condition numbers are used to study some sensitivity aspects of scalar optimization problems. The aim of this paper is to introduce a notion of condition number to study the case of a multiobjective optimization problem…
A theoretical analysis is given of the equation of motion method, due to Alben et al., to compute the eigenvalue distribution (density of states) of very large matrices. The salient feature of this method is that for matrices of the kind…
Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix…
In this paper we show that several authors have derived wrong physical conclusions from a gross misunderstanding of the exact eigenvalues and eigenfunctions of a conditionally-solvable quantum-mechanical model. It consists of an eigenvalue…
Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in…
We consider eigenvalues of a quantized cat map (i.e. hyperbolic symplectic integer matrix), cut off in phase space to include a fixed point as its only periodic orbit on the torus. We prove a simple formula for the eigenvalues on both the…