Related papers: A matrix realization of spectral bounds

We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in 2014.…

In 1972, A. J. Hoffman proved his celebrated theorem concerning the limit points of spectral radii of non-negative symmetric integral matrices less than $\sqrt{2+\sqrt{5}}$. In this paper, after giving a new version of Hoffman's theorem, we…

In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schr\"{o}dinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by…

In this paper we study the structure and give bounds for the eigenvalues of the $n\times n$ matrix, which $ij$ entry is $(i,j)^\alpha[i,j]^\beta$, where $\alpha,\beta\in\Rset$, $(i,j)$ is the greatest common divisor of $i$ and $j$ and…

The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to…

A rational matrix is a matrix-valued function $R(\lambda): \mathbb{C} \rightarrow M_p$ such that $R(\lambda) = \begin{bmatrix} r_{ij}(\lambda) \end{bmatrix}_{p\times p}$, where $r_{ij}(\lambda)$ are scalar complex rational functions in…

We describe an algorithm to compute the extremal eigenvalues and corresponding eigenvectors of a symmetric matrix by solving a sequence of Quadratic Binary Optimization problems. This algorithm is robust across many different classes of…

Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes…

We study the combinatorial and algebraic properties of Nonnegative Matrices. Our results are divided into three different categories. 1. We show a quantitative generalization of the 100 year-old Perron-Frobenius theorem, a fundamental…

This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary…

For a graph with largest normalized Laplacian eigenvalue $\lambda_N$ and (vertex) coloring number $\chi$, it is known that $\lambda_N\geq \chi/(\chi-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the…

There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random non-Hermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a…

Consider $n$ linearly independent vectors in $\mathbb{C}^n$ which form columns of a matrix $A$. The recursive evaluation of eigen directions (normalized eigenvectors) of $A$ is the solution of an eigenvalue problem of the form…

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the…

We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined…

In this paper we study the (classical) Frobenius problem, namely the problem of finding the largest integer that cannot be represented as a nonnegative integral combination of given relatively prime (strictly) positive integers (known as…

Improving upon results of Rudelson and Vershynin, we establish delocalization bounds for eigenvectors of independent-entry random matrices. In particular, we show that with high probability every eigenvector is delocalized, meaning any…

The following hypothesis was put forward by Goreinov, Tyrtyshnikov and Zamarashkin in \cite{GTZ1997}. For arbitrary real $n \times k$ matrix with orthonormal columns a sufficiently "good" $k \times k$ submatrix exists. "Good" in the sense…

We offer a new method for proving that the maximal eigenvalue of the normalized graph Laplacian of a graph with $n$ vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the…

The eigenvalue bounds obtained earlier [J. Phys. A: Math. Gen. 31 (1998) 963] for smooth transformations of the form V(x) = g(x^2) + f(1/x^2) are extended to N-dimensions. In particular a simple formula is derived which bounds the…