Related papers: A Tutorial on Matrix Perturbation Theory (using co…
A simple modification of the definition of the S-matrix is proposed. It is expected that the divergences related to nonzero self-energies are considerably milder with the modified definition than with the usual one. This conjecture is…
Chord diagrams and combinatorics of word algebras are used to model products of Dirac matrices, their traces, and contractions. A simple formula for the result of arbitrary contractions is derived, simplifying and extending an old…
The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the…
We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…
The concept of effective particles as degrees of freedom in a relativistic quantum field theory is defined using a non-perturbative renormalization group procedure for Hamiltonians. However, every candidate for a basic physical theory…
An important problem in applied dynamical systems is to compute the external forcing that provokes the largest response of a desired observable quantity. For this, we investigate the perturbation theory of Markov matrices in connection with…
These notes develop aspects of perturbation theory of matrices related to so-called diagonalisation schemes. Primary focus is on constructive tools to derive asymptotic expansions for small/large parameters of eigenvalues and…
Estimating eigenvectors and low-dimensional subspaces is of central importance for numerous problems in statistics, computer science, and applied mathematics. This paper characterizes the behavior of perturbed eigenvectors for a range of…
We present an auxiliary space theory that provides a unified framework for analyzing various iterative methods for solving linear systems that may be semidefinite. By interpreting a given iterative method for the original system as an…
We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea…
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…
This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable.…
The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent…
A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the…
We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this…
Eigenvalues are defined for any element of an algebra of observables and do not require a representation in terms of wave functions or density matrices. A systematic algebraic derivation based on moments is presented here for the harmonic…
The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the…
The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a…
Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…