Perturbation theory for the singular value decomposition. An orbitalfree quantum perturbation theory is proposed. Pt2f stands for secondorder perturbation theory with fock matrix. E2r n be symmetric matrices partitioned as a a 1 0 0 a 2. Comprehensive account of analytic perturbation theory for eigenvalues and eigenvectors. It is therefore important to be able to assess the effects of the noise on the singular values and singular vectors a problem in classical perturbation theory.
For simple eigenvalues, the theory is wellunderstood and condition numbers are available in closed form lo. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by. The second selection is attributed to, though not in the context of 2. 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. Perturbation theory for matrix equations, volume 9 1st edition. Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c perturbation theory. Condition operators, condition numbers, and condition. Most perturbation theorems say that some notion of distance between eigenstu, e.
This book is a survey of matrix perturbation theory. However, the the decomposition must be computed from a matrix consisting of both signal and noise. Algorithms and perturbation theory for matrix eigenvalue. Earlier reference, more of a functional analysis slant. 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 norms, including the theory of unitary invariant norms. I think there are some ways to put conditions on the perturbations, but this article fails to list those conditions. This paper is concerned with the hermitian definite generalized eigenvalue problem a. Yaniv shmueli tau matrix perturbation theory november 29, 2012 4 21. The main question of matrix perturbation theory is.
Perturbation theory for matrix equations volume 9 studies. The usual reference for matrix perturbation theory is the book of stewart and sun, which was written primarily with numerical issues in mind. On the status of perturbation theory 3 chaotic systems as being the lowest points in the valley. Pete stewart by including his words on this subject verbatim from his survey paper on stochastic perturbation theory 1 which has been included in the course reader random matrices ii. It is therefore important to be able to assess the eeects of the noise on the singular values and singular vectors a problem in classical perturbation theory. According to the bauerfike theorem, eigenvalue perturbation due to the perturbation e in matrix. The presented results extend those of li and li linear algebra appl.
There is an extensive theoretical literature on the influence of perturbation of coefficients of matrices on solutions of systems of equations. 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. Most of the theory concerning matrix polynomials 10,11,15 is developed for. The approach is to develop a taylor series in the perturbation which we will typically write as. Matrix perturbation theory computer science and scientific. In particular, the authors cover perturbation theory. The perturbation theory is important in applications and theoretical investigations as well. Given a nonsymmetric matrix a, we investigate the effect of perturbations on an invariant. Stewart, perturbation bounds for the definite generalized. Start with the cs decomposition and the move on to principal angles and then perturbation theory results. As each of the f i is a conserved quantity, the motion of the system is con ned to a submanifold of phase space determined by the initial values of these. In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the generalized eignevalue problem as. Masters of analytic perturbation theory and numerical linear algebra on the occasion of their 90th and 79th birthdays abstract we present.
Algorithms and perturbation theory for matrix eigenvalue problems. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. 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 enius norm. This approach has been used by stewart and sun 23 and dedieu.
Here we investigate three groups of perturbation problems which are related to computational methods of. There is a vast amount of material in matrix operator perturbation theory. Firstorder perturbation theory for eigenvalues and eigenvectors\ast anne greenbaum rencang li\ddagger michael l. Stewart and sun s matrix perturbation theory, the result is only mentioned without proof. We will assume the rst of these is the hamiltonian. I am aware of the text matrix perturbation theory by g. Pt2f secondorder perturbation theory with fock matrix. 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 norms, including the. Iv the perturbation of eigenvalues 165 1 general perturbation theorems 166 1. Sun, matrix perturbation theory 1990 yaniv shmueli tau matrix perturbation theory november 29, 2012 2 21.
Matrix perturbation theory and its applications author. A survey of componentwise perturbation theory higham. Purchase perturbation theory for matrix equations, volume 9 1st edition. Bounds on how its eigenvalues vary when a and b are perturbed by hermitian matrices are established. It is some sort of using conclusion itself to prove the. As each of the f i is a conserved quantity, the motion of the system is con ned to a submanifold of phase space determined by the initial. Introduction matrix analysis would not have developed into the vast subject it is today without the concept of representing a matrix by a single symbol. Introduction to matrix computations, academic press, new. Nonnegative matrix factorization is a computational technique of dimensional reduction of a given data to uncover the latent factors embedded in higher dimensions. Using matrix perturbation theory to update the eigenpairs. Perturbation theory for homogeneous polynomial eigenvalue. Now, we need to extend this concept to subspaces when r 1. We also consider extensions to the singular value decomposition, deriving anal. It gives the response of the density matrix upon variation of the hamiltonian by quadratically convergent recursions based on perturbed.
The second part of this dissertation explores eigenvalue perturbation theory. An important result in eigenvalue perturbation theory li and li, 2005 is. This type of analysis was popularized by lord rayleigh, in his investigation of harmonic. A critical feature of the technique is a middle step that breaks the problem into solvable and perturbation parts. Perturbation theories is in many cases the only theoretical technique that we have to handle various complex systems quantum and classical. This pro vides us with a new form ulation to the least squares estimation problem in whic h uncertain t y. Perturbation theory the unsymmetric eigenvalue problem just as the problem of solving a system of linear equations ax b can be sensitive to perturbations in the data, the problem of computing the eigenvalues of a matrix can also be sensitive to perturbations in the matrix. Firstorder perturbation theory for eigenvalues and. Besides being of theoretical interest, perturbation theory is a useful tool that plays important roles in many applications.
Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Table of contents for matrix perturbation theory g. Its a wellknown result and it isnt a difficult one. I think most authors would feel no need to include any proofs in their books. Quadratic stark effect degenerate perturbation theory let us, rather naively, investigate the stark effect in an excited i. Timedependent perturbation theory so far, we have focused on quantum mechanics of systems described by hamiltonians that are timeindependent. 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.
It was noted in stewart and sun matrix perturbation theory. Density matrix perturbation theory of magnetooptical. It is worth comparing our result with that of stewart and sun 146, cor. Similarly, perturbation theory would not be such a. At every cosmologically interesting scale, it aptly and succesfully describes the situation in the early eons after the decoupling of radiation and matter at recombination. If you are working with matrices, you will want a copy. This permits the computation of statistics estimating the variation in the perturbed quantity.
Unlike traditional matrix decomposition methods such as svd and full rank decomposition, the nonnegativity constraint imposed by nmf is useful for learning partbased representations. 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. Comprehensive account of perturbation theory for linear algebra problems. Problems and results in matrix perturbation theory springerlink. Perturbation theory is used to estimate the energies and wave functions for a quantum system described by a potential which is only slightly di. This book is a comprehensive survey of matrix perturbation theory, a topic of interest to numerical analysts, statisticians, physical scientists, and engineers. Nondegenerate perturbation theory was used in this case even though the 0 are highly degenerate. Princeton university library one washington road princeton, nj 085442098 usa 609 2581470. The classical perturbation theory for matrix eigenvalue and singular value prob lems provides.
B for block diagonal matrices adiaga11,a22 and bdiagb11,b22. The perturbed quantity is approximated by a rstorder perturbation expansion, in which the perturbation is assumed to be random. Perturbation theory for the eigenvalues of factorised symmetric matrices, linear algebra. For eigenvectors and invariant subspaces of multiple eigenvalues, much work has been done by stewart. It was noted by stewart and sun matrix perturbation theory, academic press, boston, 1990 that different copies of a multiple eigenvalue may exhibit quite different sensitivities towards perturbations. As we know a little kick let a ball at the top going far down, without stopping inbetween. 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. What is the smallest p ossible erturbation of a matrix that causes it to lose rank. I have dozens of matrix theory books, and this is one of the best. We need to first prove that if the matrix perturbations are small, the changes in eigenvalues and eigenvectors will be small too.
It may have been proven somewhere, but in the relevant books that ive read e. In this paper we survey the perturbation theory of the singular value decomposition. Bibliographic record and links to related information available from the library of congress catalog information from electronic data provided by the publisher. Let a be a matrix and let f be a matrix valued function of a.
Perturbation theory for the singular value decomposition 1990. These bounds are generally of linear order with respect to the perturbations in the. Matrix perturbation theory, academic press, san diego. Rubin december 2, 2010 lecture 3241 november 10 december 3, 2010 1 stationary state perturbation theory 1. The symmetric eigenvalue problem, society for industrial and applied mathematics. This theory can be used for a number of interesting subjects in lca. Katos perturbation theory for linear operators with awe and bewilderment. A note on eigenvalues of perturbed hermitian matrices request pdf. In this book a general perturbation theory for matrix algebraic equations is presented. The problem for defective multiple eigenvalues has not been considered until recently 3.
Harcourt bruce jovanovich, publishers boston san diego new york london sydney tokyo toronto. John lipor assistant professor, portland state university. In such cases, time dependence of wavefunction developed through. Good references on perturbation theory are stewart and sun 1990. In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the. Such methods can accelerate algorithms like pagerank and hit that use the stationary distribution values as rating scores. This is useful for studying how sensitive the original systems eigenvectors and eigenvalues are to changes in the system. How does the eigenstructure of a matrix changes when its elements are subjected to a perturbation. How is secondorder perturbation theory with fock matrix abbreviated. Bibliographic record and links to related information available from the library of congress catalog. Both a and b are hermitian, and b is positive definite. Because changes in the m matrix can even make the problem illdefined, see chapter vi in stewart, g. Niklasson, matt challacombe submitted on 26 nov 2003 v1, last revised 3 dec 2003 this version, v2.
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