Related papers: Perturbation theory for the matrix square root and…
Let A and E be Hermitian self-adjoint matrices, where A is fixed and E a small perturbation. We study how the eigenvalues and eigenvectors of A+E depend on E, with the aim of obtaining first order formulas (and when possible also second…
We consider "spectral" matrix-functions for Hermitian matrices, where the novelty is that the function applied to the spectrum is allowed to be a vector-field rather than a scalar function (a.k.a isotropic matrix functions). We prove first…
We provide a systematic formula, in terms of integer partitions, that generates perturbation theory explicitly at an arbitrary order. Our approach naturally includes an infinite number of perturbations and uses a single matrix equation that…
The first well founded perturbation theory for classical solid systems is presented. Theoretical approaches to thermodynamic and structural properties of the hard-sphere solid provide us with the reference system. The traditional…
In this paper, we establish a useful set of formulae for the $\sin\Theta$ distance between the original and the perturbed singular subspaces. These formulae explicitly show that how the perturbation of the original matrix propagates into…
We report lowest-order series expansions for primary matrix functions of quantum states based on a perturbation theory for functions of linear operators. Our theory enables efficient computation of functions of perturbed quantum states that…
In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.
Consider a matrix function f defined for Hermitian matrices. The purpose of this paper is two-fold. We derive new results for the absolute structured condition number of the matrix function and we derive new bounds for the perturbation…
We discuss the effect of perturbations on the ground rings of $c=1$ string theory at the various compactification radii defining the $A_N$ points of the moduli space. We argue that perturbations by plus-type moduli define ground varieties…
We present theory for general partial derivatives of matrix functions on the form $f(A(x))$ where $A(x)$ is a matrix path of several variables ($x=(x_1,\dots,x_j)$). Building on results by Mathias [SIAM J. Matrix Anal. Appl., 17 (1996), pp.…
This report discusses two new ideas for using perturbation methods to solve the time-independent Schr\"odinger equation. The first concept begins with rewriting the perturbation equations in a form that is closely related to matrix…
Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In…
We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad…
We prove quadratic eigenvalue perturbation bounds for generalized Hermitian eigenvalue problems. The bounds are proportional to the square of the norm of the perturbation matrices divided by the gap between the spectrums. Using the results…
We have obtained the perturbative expressions up to sixth order for the energy of the bound state in a one dimensional, arbitrarily weak, short range finite well, applying a method originally developed by Gat and Rosenstein Ref. [3]. The…
The Riemann-Hilbert problem associated with the integrable PDE is used as a nonlinear transformation of the nearly integrable PDE to the spectral space. The temporal evolution of the spectral data is derived with account for arbitrary…
We implement Lie transform perturbation theory to second order for the planar spin-orbit problem. The perturbation parameter is the asphericity of the body, with the orbital eccentricity entering as an additional parameter. We study first…
Combined perturbation bounds are presented for eigenvalues and eigenspaces of Hermitian matrices or singular values and singular subspaces of general matrices. The bounds are derived based on the smooth decompositions and elementary…
This paper establishes a variant of Stewart's theorem (Theorem~6.4 of Stewart, {\em SIAM Rev.}, 15:727--764, 1973) for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both…
In this paper, we study the perturbation of the extreme singular values of a matrix in the particular case where it is obtained after appending an arbitrary column vector. Such results have many applications in bifurcation theory, signal…