Related papers: A Test Matrix for an Inverse Eigenvalue Problem

In this short note, we present a novel method for computing exact lower and upper bounds of eigenvalues of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and…

In this work, the real nonnegative inverse eigenvalue problem is solved for a particular class of permutative matrix. The necessary and sufficient condition there is also shown to be sufficient for the symmetric nonnegative inverse…

A $2n\times 2n$ real matrix $A$ is said to be a Hamiltonian matrix if $A^{T}J+JA=0$, where $J=\left( \begin{array}{cc} 0 & I_{n} \\ -I_{n} & 0\\ \end{array} \right)$. Hamiltonian matrices appear in many areas of applications, such as linear…

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be…

In this work, we study the numerical solution of inverse eigenvalue problems from a machine learning perspective. Two different problems are considered: the inverse Strum-Liouville eigenvalue problem for symmetric potentials and the inverse…

We present the explicit inverse of a class of symmetric tridiagonal matrices which is almost Toeplitz, except that the first and last diagonal elements are different from the rest. This class of tridiagonal matrices are of special interest…

We show that if a list of nonzero complex numbers $\sigma=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is the nonzero spectrum of a diagonalizable nonnegative matrix, then $\sigma$ is the nonzero spectrum of a diagonalizable nonnegative matrix…

In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…

We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of…

This paper has solved the inverse eigenvalue problem for "fixed-free" mass-chain systems with inerters. It is well known that for a spring-mass system wherein the adjacent masses are linked through a spring, the natural frequency assignment…

The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph $G$. In this paper, we refer to the $i$-nullity pair of a matrix…

For given k distinct complex conjugate pairs, l distinct real numbers, and a given graph G on 2k+l vertices with a matching of size at least k, we will show that there is a real matrix whose eigenvalues are the given numbers and its graph…

Consider the Jacobi operators $\cJ$ given by $(\cJ y)_n=a_ny_{n+1}+b_ny_n+a_{n-1}^*y_{n-1}$, $y_n\in \C^m$ (here $y_0=y_{p+1}=0$), where $b_n=b_n^*$ and $a_n:\det a_n\ne 0$ are the sequences of $m\ts m$ matrices, $n=1,..,p$. We study two…

Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal matrix with $-1$'s on the sub-diagonal, $1$'s on the super-diagonal, $-1$ in the $(1,1)$ entry, $1$ in the $(n,n)$ entry and zeros elsewhere. We find the eigen-pairs…

An $n\times n$ matrix $C$ is said to be {\it centrosymmetric} if it satisfies the relation $JCJ=C$, where $J$ is the $n\times n$ counteridentity matrix. Centrosymmetric matrices have a rich eigenstructure that has been studied extensively…

Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$ where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0 \leq…

This paper examines the properties of real symmetric square matrices with a constant value for the main diagonal elements and another constant value for all off-diagonal elements. This matrix form is a simple subclass of circulant matrices,…

We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse. The techniques we use are related with the…

We consider the inverse eigenvalue problem for entanglement witnesses, which asks for a characterization of their possible spectra (or equivalently, of the possible spectra resulting from positive linear maps of matrices). We completely…

Building on previous work that provided analytical solutions to generalised matrix eigenvalue problems arising from numerical discretisations, this paper develops exact eigenvalues and eigenvectors for a broader class of $n$-dimensional…