Related papers: The Nearest Hermitian Inverse Eigenvalue Problem S…
We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix $A$. Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to…
We discuss the eigenvalue problem for 2x2 and 3x3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-real eigenvalues.
In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $\lambda^2 M x + \lambda C x + K x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices…
We first consider the problem of approximating a few eigenvalues of a rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational part of the rational matrix-valued function is expressed in the…
The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix…
We propose and study an algorithm for computing a nearest passive system to a given non-passive linear time-invariant system (with much freedom in the choice of the metric defining `nearest', which may be restricted to structured…
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…
In this work we consider a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axis. More precisely, we…
We construct six multi-parameter families of Hermitian quasi-exactly solvable matrix Schroedinger operators in one variable. The method for finding these operators relies heavily upon a special representation of the Lie algebra o(2,2) whose…
We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear…
In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential…
Non-smoothness at optimal points is a common phenomenon in many eigenvalue optimization problems. We consider two recent algorithms to minimize the largest eigenvalue of a Hermitian matrix dependent on one parameter, both proven to be…
We consider the class of non-Hermitian operators represented by infinite tridiagonal matrices, selfadjoint in an indefinite inner product space with one negative square. We approximate them with their finite truncations. Both infinite and…
The inverse eigenvalue problem for real symmetric matrices of the form 0 0 0 . 0 0 * 0 0 0 . 0 * * 0 0 0 . * * 0 . . . . . . . 0 0 * . 0 0 0 0 * * . 0 0 0 * * 0 . 0 0 0 is solved. The solution is shown to be unique. The problem is also…
This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued function depending on its parameters analytically in a box. We describe how the analytical…
We derive a sufficient condition for a Hermitian $N \times N$ matrix $A$ to have at least $m$ eigenvalues (counting multiplicities) in the interval $(-\epsilon, \epsilon)$. This condition is expressed in terms of the existence of a…
We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries…
Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This…
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of…
Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based…