Related papers: Min-plus methods in eigenvalue perturbation theory…

Concatenating matrices is a common technique for uncovering shared structures in data through singular value decomposition (SVD) and low-rank approximations. The fundamental question arises: How does the singular value spectrum of the…

Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of an operator where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several…

In this paper, the discontinuous Petrov--Galerkin approximation of the Laplace eigenvalue problem is discussed. We consider in particular the primal and ultra weak formulations of the problem and prove the convergence together with a priori…

In this paper, we propose an unconstrained framework for eigenvalue problems in both discrete and continuous settings. We begin our discussion to solve a generalized eigenvalue problem $A{\bf x} = \lambda B{\bf x}$ with two $N\times N$ real…

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue''…

The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. While there exists a large literature on the existence of solutions for special settings, there are only few numerical solution methods…

We give a new proof for an equality of certain max-min and min-max approximation problems involving normal matrices. The previously published proofs of this equality apply tools from matrix theory, (analytic) optimization theory and…

In a previous paper (J. Phys. A 36, 11807 (2003)), we introduced the `asymptotic iteration method' for solving second-order homogeneous linear differential equations. In this paper, we study perturbed problems in quantum mechanics and we…

We introduce a max-plus analogue of the Petrov-Galerkin finite element method to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation. We show that the error in the sup norm…

We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is…

Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the…

Analytical tools to $K$-theory; namely, self-stabilization of rapidly decreasing matrices, linearization of cyclic loops, and the contractibility of the pointed stable Toeplitz algebra are discussed in terms of concrete formulas. Adaptation…

We consider the smallest eigenvalues of perturbed Hermitian operators with zero modes, either topological or system specific. To leading order for small generic perturbation we show that the corresponding eigenvalues broaden to a Gaussian…

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $\mathbb{R}^N$. We prove that the symmetric functions of the eigenvalues depend real analytically…

Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C <= A + B are given by the same inequalities as in Klyachko's theorem for the case where C = A…

Eigenvalue and eigenvector perturbation theory is a fundamental topic in several disciplines, including numerical linear algebra, quantum physics, and related fields. The central problem is to understand how the eigenvalues and eigenvectors…

This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in "User-friendly tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both upper…

We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of…

We provide first order perturbation formulas for the matrix square root (in the positive semi-definite case) and the matrix modulus (in the general case). The results are new for singular matrices, and extend previously known Fr\'{e}chet…

This paper studies the problem of selecting a submatrix of a positive definite matrix in order to achieve a desired bound on the smallest eigenvalue of the submatrix. Maximizing this smallest eigenvalue has applications to selecting input…