The second part of this dissertation explores eigenvalue 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. Its a wellknown result and it isnt a difficult one. Iv the perturbation of eigenvalues 165 1 general perturbation theorems 166 1. Algorithms and perturbation theory for matrix eigenvalue problems. This type of analysis was popularized by lord rayleigh, in his investigation of harmonic. John lipor assistant professor, portland state university. Matrix perturbation theory and its applications author. Condition operators, condition numbers, and condition. Firstorder perturbation theory for eigenvalues and eigenvectors\ast anne greenbaum rencang li\ddagger michael l. B for block diagonal matrices adiaga11,a22 and bdiagb11,b22. Start with the cs decomposition and the move on to principal angles and then perturbation theory results. Comprehensive account of perturbation theory for linear algebra problems. Matrix perturbation theory computer science and scientific.
Here we investigate three groups of perturbation problems which are related to computational methods of. Bibliographic record and links to related information available from the library of congress catalog information from electronic data provided by the publisher. How does the eigenstructure of a matrix changes when its elements are subjected to a perturbation. As we know a little kick let a ball at the top going far down, without stopping inbetween. This book is a survey of matrix perturbation theory. Because changes in the m matrix can even make the problem illdefined, see chapter vi in stewart, g. It may have been proven somewhere, but in the relevant books that ive read e. Perturbation theory for the singular value decomposition 1990. The presented results extend those of li and li linear algebra appl. Algorithms and perturbation theory for matrix eigenvalue.
Sun, matrix perturbation theory 1990 yaniv shmueli tau matrix perturbation theory november 29, 2012 2 21. Matrix perturbation theory, academic press, san diego. In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues. This book is a comprehensive survey of matrix perturbation theory, a topic of interest to numerical analysts, statisticians, physical scientists, and engineers. For simple eigenvalues, the theory is wellunderstood and condition numbers are available in closed form lo. For example, it is frequently employed in the stability analysis of.
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 problem for defective multiple eigenvalues has not been considered until recently 3. Both a and b are hermitian, and b is positive definite. In this book a general perturbation theory for matrix algebraic equations is presented. In such cases, time dependence of wavefunction developed through.
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. 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. A survey of componentwise perturbation theory higham. It gives the response of the density matrix upon variation of the hamiltonian by quadratically convergent recursions based on perturbed. Perturbation theory for the singular value decomposition. The usual reference for matrix perturbation theory is the book of stewart and sun, which was written primarily with numerical issues in mind. Perturbation theory for homogeneous polynomial eigenvalue. There is an extensive theoretical literature on the influence of perturbation of coefficients of matrices on solutions of systems of equations. Perturbation theory for the eigenvalues of factorised symmetric matrices, linear algebra. Pt2f stands for secondorder perturbation theory with fock matrix. It is some sort of using conclusion itself to prove the. Perturbation theory and derivatives of matrix eigensystems article pdf available in applied mathematics letters 14.
Masters of analytic perturbation theory and numerical linear algebra on the occasion of their 90th and 79th birthdays abstract we present. The perturbation theory is important in applications and theoretical investigations as well. An orbitalfree quantum perturbation theory is proposed. I have dozens of matrix theory books, and this is one of the best. A note on eigenvalues of perturbed hermitian matrices request pdf. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by. 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. Perturbation theory for matrix equations volume 9 studies. 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. The second selection is attributed to, though not in the context of 2. The symmetric eigenvalue problem, society for industrial and applied mathematics.
Using matrix perturbation theory to update the eigenpairs. The perturbed quantity is approximated by a rstorder perturbation expansion, in which the perturbation is assumed to be random. 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. 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. Similarly, perturbation theory would not be such a. In this paper classical matrix perturbation theory is approached from a probabilistic point of view. We also consider extensions to the singular value decomposition, deriving anal. Now, we need to extend this concept to subspaces when r 1. This permits the computation of statistics estimating the variation in the perturbed quantity. Most of the theory concerning matrix polynomials 10,11,15 is developed for. Quadratic stark effect degenerate perturbation theory let us, rather naively, investigate the stark effect in an excited i.
It was noted in stewart and sun matrix perturbation theory. 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. An important result in eigenvalue perturbation theory li and li, 2005 is. Nondegenerate perturbation theory was used in this case even though the 0 are highly degenerate. Bibliographic record and links to related information available from the library of congress catalog.
In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the generalized eignevalue problem as. Pt2f secondorder perturbation theory with fock matrix. Table of contents for matrix perturbation theory g. If you are working with matrices, you will want a copy.
I think most authors would feel no need to include any proofs in their books. 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. Density matrix perturbation theory of magnetooptical. The classical perturbation theory for matrix eigenvalue and singular value prob lems provides. Stewart and sun s matrix perturbation theory, the result is only mentioned without proof. There is a vast amount of material in matrix operator perturbation theory. What is the smallest p ossible erturbation of a matrix that causes it to lose rank. Comprehensive account of analytic perturbation theory for eigenvalues and eigenvectors.
Bounds on how its eigenvalues vary when a and b are perturbed by hermitian matrices are established. Such methods can accelerate algorithms like pagerank and hit that use the stationary distribution values as rating scores. Most perturbation theorems say that some notion of distance between eigenstu, e. Timedependent perturbation theory so far, we have focused on quantum mechanics of systems described by hamiltonians that are timeindependent. The approach is to develop a taylor series in the perturbation which we will typically write as. Pete stewart, masters of analytic perturbation theory and numerical linear algebra on the occasion of their 90th and 80th birthdays abstract. How is secondorder perturbation theory with fock matrix abbreviated. Perturbation theory for matrix equations, volume 9 1st edition. We need to first prove that if the matrix perturbations are small, the changes in eigenvalues and eigenvectors will be small too. It is worth comparing our result with that of stewart and sun 146, cor. 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.
This approach has been used by stewart and sun 23 and dedieu. Perturbation theory is used to estimate the energies and wave functions for a quantum system described by a potential which is only slightly di. Given a nonsymmetric matrix a, we investigate the effect of perturbations on an invariant. 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. Earlier reference, more of a functional analysis slant. Niklasson, matt challacombe submitted on 26 nov 2003 v1, last revised 3 dec 2003 this version, v2. We will assume the rst of these is the hamiltonian. 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. This theory can be used for a number of interesting subjects in lca. However, the the decomposition must be computed from a matrix consisting of both signal and noise. In this paper we survey the perturbation theory of the singular value decomposition. A critical feature of the technique is a middle step that breaks the problem into solvable and perturbation parts. Katos perturbation theory for linear operators with awe and bewilderment. These bounds are generally of linear order with respect to the perturbations in the.
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. This is useful for studying how sensitive the original systems eigenvectors and eigenvalues are to changes in the system. Introduction to matrix computations, academic press, new. In particular, the authors cover perturbation theory. E2r n be symmetric matrices partitioned as a a 1 0 0 a 2. Firstorder perturbation theory for eigenvalues and. Princeton university library one washington road princeton, nj 085442098 usa 609 2581470. 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. Nonnegative matrix factorization is a computational technique of dimensional reduction of a given data to uncover the latent factors embedded in higher dimensions. Yaniv shmueli tau matrix perturbation theory november 29, 2012 4 21. 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. Harcourt bruce jovanovich, publishers boston san diego new york london sydney tokyo toronto. I am aware of the text matrix perturbation theory by g.
Pt2f is defined as secondorder perturbation theory with fock matrix very rarely. Stewart, perturbation bounds for the definite generalized. Problems and results in matrix perturbation theory springerlink. I think there are some ways to put conditions on the perturbations, but this article fails to list those conditions. Purchase perturbation theory for matrix equations, volume 9 1st edition. For eigenvectors and invariant subspaces of multiple eigenvalues, much work has been done by stewart.
In particular, the authors cover perturbation theory of linear systems and least square problems, the eignevalue problem, and the. Besides being of theoretical interest, perturbation theory is a useful tool that plays important roles in many applications. Rubin december 2, 2010 lecture 3241 november 10 december 3, 2010 1 stationary state perturbation theory 1. Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c perturbation theory.
265 1160 1640 1327 648 1546 1347 776 217 164 626 131 568 370 118 774 936 1207 109 320 1324 463 1369 326 48 666 938 733 916 496 806