An optimal differential sensitivity approach to pole assignment for structurally perturbed systems hei ka tam, hei ka tam. Harcourt bruce jovanovich, publishers boston san diego new york london sydney tokyo toronto. Perturbation analyses of matrix factorizations are not only important in their own right, but also useful in many applications, e. Perturbation of the svd in the presence of small singular values perturbation of the svd in the presence of small singular values stewart, michael 20061101 00. In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the generalized eignevalue problem as wellas a complete treatment of vector and matrix. Stewart and jiguang sunv rajendra bhatia indian statistical institute new delhi110016, india in the seventies one could hear many analysts talk of the first two chapters of t. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The book is designed to meet many different needs, and because it is mathematically rigorous, it may be used by students of pure and applied mathematics. Stewart is a member of the national academy of engineering and has received numerous additional honors, including the bauer prize. Matrix perturbation theory computer science and scientific. Sun, 1990, matrix perturbation theory, academic press. The optimal perturbation bounds of the moorepenrose.
W e discuss t w o cases next, with p erturbations measured in the 2norm, and then discuss measuremen t of p erturbations f rob. Robust design for dynamic system under model uncertainty. Stewart computer science department institute for advanced computer studies university of maryland college park, maryland jiguang sun computing center ofthe chinese academy of sciences beijing, china academic press, inc. Nonnegative matrix factorization is a computational technique of dimensional reduction of a given data to uncover the latent factors embedded in higher dimensions. In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the. Buy matrix perturbation theory computer science and scientific computing on. The package provides routines for first and secondorder rayleighschroedinger perturbation theory in a matrix hamiltonian formalism, i. This book is a survey of matrix perturbation theory.
Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. The perturbed quantity is approximated by a rstorder perturbation expansion, in which the perturbation is assumed to be random. Roundingerror and perturbation bounds for the indefinite. Lapackstyle algorithms and software for solving the. Earlier reference, more of a functional analysis slant. Only a few qm systems that can be solved exactly hydrogen atomsimplified, harmonic oscillator, infinitefinite well. I have dozens of matrix theory books, and this is one of the best.
Keywords linear algebra lunch break matrix algorithm numerical linear algebra concise explanation. Specifically, block matrix level 3 blas, as in lapack codes algorithms are implemented for solving generalized sylvester equations arlbc, drlef, where r and l are unknown. Yaniv shmueli tau matrix perturbation theory november 29, 2012 4 21. This approximation could make the traditional design method less effective due to the existence of model uncertainty. An optimal differential sensitivity approach to pole. A tutorial on matrix perturbation theory using compact matrix.
By way of theory, the author has chosen to discuss the theory of norms and perturbation theory for linear systems and for the algebraic eigenvalue problem. In particular, the authors cover perturbation theory of linear systems and least square problems, the eigenvalue problem, and the generalized eigenvalue problem as well as a treatment of vector and matrix norms, including the theory of unitary invariant norms. This permits the computation of statistics estimating the variation in the. Department of mechanical engineering, university of hong kong, pokfulam road, hong kong.
There is a vast amount of material in matrix operator perturbation theory. Unlike traditional matrix decomposition methods such as svd and full rank decomposition, the nonnegativity constraint imposed by nmf is useful for learning partbased representations. A large number of examples, tables and figures is included in order to illustrate the perturbation. The daviskahan theorem is used in the analysis of many statistical procedures to bound the distance between subspaces spanned by population eigenvectors and their sample versions.
Matrix perturbation theory 1990 edition open library. The idea is to start with a simple system for which a mathematical solution is known, and add an additional perturbing hamiltonian representing a weak disturbance to the system. The perturbed quantity is approximated by a first order perturbation expansion, in which the. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be. Katos perturbation theory for linear operators with awe and bewilderment. A tutorial on matrix perturbation theory using compact matrix notation. In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. In this paper, a novel stabilitybased approach is proposed to design the system ensuring robust stability under model uncertainty. Time dependent perturbation theory 1 time dependent perturbation theory. In practice, the matrix b is often considered as the perturbed matrix of a, and hence based on the previous results, the additive perturbation bounds for the 1, i and 1, 2, iinverses and. What is the smallest p ossible erturbation of a matrix that causes it to lose rank. Local and nonlocal perturbation bounds are derived for general types of matrix equations as well as for the most important equations arising in linear algebra and control theory.
Niklasson, matt challacombe submitted on 26 nov 2003 v1, last revised 3 dec 2003 this version, v2. Siam journal on scientific and statistical computing. Fortyfour of stewart s most influential research papers in two subject areas. This factorization can be used for accurate computation of eigenvalues of the hermitian matrix ag. Stewart, jiguang sun this book is a comprehensive survey of matrix perturbation theory, a topic of interest to numerical analysts, statisticians, physical scientists, and engineers. A critical feature of the technique is a middle step that breaks the problem into solvable and perturbation parts. Matrix perturbation theory and its applications author. The perturbed quantity is approximated by a rstorder perturbation expansion, in which the perturbation. This permits the computation of statistics estimating the variation in the perturbed quantity. Indefinite qr factorization is a generalization of the wellknown qr factorization, where q is a unitary matrix with respect to the given indefinite inner product matrix j. Despite many empirical successes of spectral clustering methods algorithms that cluster points using eigenvectors of matrices derived from the distances between the points there are several unresolved issues. Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c perturbation theory. Perturbation bounds for the definite generaked eigenvalue. The first part of the paper gives subspace bounds that are closely.
First, there is a wide variety of algorithms that use the eigenvectors in. However the vast majority of systems in nature cannot be solved exactly, and we need. The preceding seven chapters of this commentary had outlined some of stewart s important contributions to matrix algorithms and matrix perturbation theory. Introduction to matrix computations computer science and.