DOCSLIB.ORG

The Eigenvalue Problem: Perturbation Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Eigenvalue Problem: Perturbation Theory Jim Lambers MAT 610 Summer Session 2009-10 Lecture 13 Notes These notes correspond to Sections 7.2 and 8.1 in the text. The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem Just as the problem of solving a system of linear equations Ax = b can be sensitive to pertur- bations in the data, the problem of computing the eigenvalues of a matrix can also be sensitive to perturbations in the matrix. We will now obtain some results concerning the extent of this sensitivity. Suppose that A is obtained by perturbing a diagonal matrix D by a matrix F whose diagonal entries are zero; that is, A = D + F . If is an eigenvalue of A with corresponding eigenvector x, then we have (D − I)x + F x = 0: If is not equal to any of the diagonal entries of A, then D − I is nonsingular and we have x = −(D − I)−1F x: Taking 1-norms of both sides, we obtain −1 −1 kxk1 = k(D − I) F xk1 ≤ k(D − I) F k1kxk1; which yields n −1 X jfijj k(D − I) F k1 = max ≥ 1: 1≤i≤n jdii − j j=1;j6=i It follows that for some i, 1 ≤ i ≤ n, satisfies n X jdii − j ≤ jfijj: j=1;j6=i That is, lies within one of the Gerschgorin circles in the complex plane, that has center aii and radius n X ri = jaijj: j=1;j6=i 1 This is result is known as the Gerschgorin Circle Theorem. Example The eigenvalues of the matrix 2 −5 −1 1 3 A = 4 −2 2 −1 5 1 −3 7 are (A) = f6:4971; 2:7930; −5:2902g: The Gerschgorin disks are D1 = fz 2 C j jz − 7j ≤ 4g;D2 = fz 2 C j jz − 2j ≤ 3g;D3 = fz 2 C j jz + 5j ≤ 2g: We see that each disk contains one eigenvalue. 2 It is important to note that while there are n eigenvalues and n Gerschgorin disks, it is not necessarily true that each disk contains an eigenvalue. The Gerschgorin Circle Theorem only states that all of the eigenvalues are contained within the union of the disks. Another useful sensitivity result that applies to diagonalizable matrices is the Bauer-Fike The- orem, which states that if −1 X AX = diag(1; : : : ; n); and is an eigenvalue of a perturbed matrix A + E, then min j − j ≤ p(X)kEkp: 2(A) That is, is within p(X)kEkp of an eigenvalue of A. It follows that if A is \nearly non- diagonalizable", which can be the case if eigenvectors are nearly linearly dependent, then a small perturbation in A could still cause a large change in the eigenvalues. It would be desirable to have a concrete measure of the sensitivity of an eigenvalue, just as we have the condition number that measures the sensitivity of a system of linear equations. To that end, we assume that is a simple eigenvalue of an n × n matrix A that has Jordan canonical form −1 J = X AX. Then, = Jii for some i, and xi, the ith column of X, is a corresponding right eigenvector. −H −1 H If we define Y = X = (X ) , then yi is a left eigenvector of A corresponding to . From Y H X = I, it follows that yH x = 1. We now let A, , and x be functions of a parameter that satisfy A()x() = ()x();A() = A + F; kF k2 = 1: Differentiating with respect to , and evaluating at = 0, yields F x + Ax0(0) = x0(0) + 0(0)x: 2 Taking the inner product of both sides with y yields yH F x + yH Ax0(0) = yH x0(0) + 0(0)yH x: Because y is a left eigenvector corresponding to , and yH x = 1, we have yH F x + yH x0(0) = yH x0(0) + 0(0): We conclude that 0 H j (0)j = jy F xj ≤ kyk2kF k2kxk2 ≤ kyk2kxk2: However, because , the angle between x and y, is given by yH x 1 cos = = ; kyk2kxk2 kyk2kxk2 it follows that 1 j0(0)j ≤ : j cos j We define 1=j cos j to be the condition number of the simple eigenvalue . We require to be simple because otherwise, the angle between the left and right eigenvectors is not unique, because the eigenvectors themselves are not unique. It should be noted that the condition number is also defined by 1=jyH xj, where x and y are normalized so that kxk2 = kyk2 = 1, but either way, the condition number is equal to 1=j cos j. The interpretation of the condition number is that an O() perturbation in A can cause an O(/j cos j) perturbation in the eigenvalue . Therefore, if x and y are nearly orthogonal, a large change in the eigenvalue can occur. Furthermore, if the condition number is large, then A is close to a matrix with a multiple eigenvalue. Example The matrix 2 3:1482 −0:2017 −0:5363 3 A = 4 0:4196 0:5171 1:0888 5 0:3658 −1:7169 3:3361 has a simple eigenvalue = 1:9833 with left and right eigenvectors x = 0:4150 0:6160 0:6696 T ; y = −7:9435 83:0701 −70:0066 T H such that y x = 1. It follows that the condition number of this eigenvalue is kxk2kyk2 = 108:925. In fact, the nearby matrix 2 3:1477 −0:2023 −0:5366 3 B = 4 0:4187 0:5169 1:0883 5 0:3654 −1:7176 3:3354 3 has a double eigenvalue that is equal to 2. 2 We now consider the sensitivity of repeated eigenvalues. First, it is important to note that while the eigenvalues of a matrix A are continuous functions of the entries of A, they are not necessarily differentiable functions of the entries. To see this, we consider the matrix 1 a A = ; 1 where a > 0. Computing its characteristic polynomial det(A − I) = 2 − 2 + 1 − a p and computings its roots yields the eigenvalues = 1 ± a. Differentiating these eigenvalues with respect to yields d ra = ± ; d which is undefined at = 0. In general, an O() perturbation in A causes an O(1=p) perturbation in an eigenvalue associated with a p×p Jordan block, meaning that the \more defective" an eigenvalue is, the more sensitive it is. We now consider the sensitivity of eigenvectors, or, more generally, invariant subspaces of a matrix A, such as a subspace spanned by the first k Schur vectors, which are the first k columns in a matrix Q such that QH AQ is upper triangular. Suppose that an n × n matrix A has the Schur decomposition H T11 T12 A = QT Q ;Q = Q1 Q2 ;T = ; 0 T22 where Q1 is n × r and T11 is r × r. We define the separation between the matrices T11 and T22 by kT11X − XT22kF sep(T11;T22) = min : X6=0 kXkF It can be shown that an O() perturbation in A causes a O(/sep(T11;T22)) perturbation in the invariant subspace Q1. We now consider the case where r = 1, meaning that Q1 is actually a vector q1, that is also an eigenvector, and T11 is the corresponding eigenvalue, . Then, we have kX − XT22kF sep(, T22) = min X6=0 kXkF H = min ky (T22 − I)k2 kyk2=1 H = min k(T22 − I) yk2 kyk2=1 H = min((T22 − I ) ) = min(T22 − I); 4 since the Frobenius norm of a vector is equivalent to the vector 2-norm. Because the smallest singular value indicates the distance to a singular matrix, sep(, T22) provides a measure of the separation of from the other eigenvalues of A. It follows that eigenvectors are more sensitive to perturbation if the corresponding eigenvalues are clustered near one another. That is, eigenvectors associated with nearby eigenvalues are \wobbly". It should be emphasized that there is no direct relationship between the sensitivity of an eigen- value and the sensitivity of its corresponding invariant subspace. The sensitivity of a simple eigen- value depends on the angle between its left and right eigenvectors, while the sensitivity of an invariant subspace depends on the clustering of the eigenvalues. Therefore, a sensitive eigenvalue, that is nearly defective, can be associated with an insensitive invariant subspace, if it is distant from other eigenvalues, while an insensitive eigenvalue can have a sensitive invariant subspace if it is very close to other eigenvalues. The Symmetric Eigenvalue Problem In the symmetric case, the Gerschgorin circles become Gerschgorin intervals, because the eigenval- ues of a symmetric matrix are real. Example The eigenvalues of the 3 × 3 symmetric matrix 2 −10 −3 2 3 A = 4 −3 4 −2 5 2 −2 14 are (A) = f14:6515; 4:0638; −10:7153g: The Gerschgorin intervals are D1 = fx 2 R j jx − 14j ≤ 4g;D2 = fx 2 R j jx − 4j ≤ 5g;D3 = fx 2 R j jx + 10j ≤ 5g: We see that each intervals contains one eigenvalue. 2 The characterization of the eigenvalues of a symmetric matrix as constrained maxima of the Rayleight quotient lead to the following results about the eigenvalues of a perturbed symmetric matrix. As the eigenvalues are real, and therefore can be ordered, we denote by i(A) the ith largest eigenvalue of A. Theorem (Wielandt-Hoffman) If A and A + E are n × n symmetric matrices, then n X 2 2 (i(A + E) − i(A)) ≤ kEkF : i=1 It is also possible to bound the distance between individual eigenvalues of A and A + E. 5 Theorem If A and A + E are n × n symmetric matrices, then n(E) ≤ k(A + E) − k(A) ≤ 1(E): Furthermore, jk(A + E) − k(A)j ≤ kEk2: The second inequality in the above theorem follows directly from the first, as the 2-norm of the symmetric matrix E, being equal to its spectral radius, must be equal to the larger of the absolute value of 1(E) or n(E).
Load more

Recommended publications
  • Structured Eigenvalue Condition Numbers and Linearizations for Matrix Polynomials
    ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Eidgen¨ossische Ecole polytechnique f´ed´erale de Zurich ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Technische Hochschule Politecnico federale di Zurigo ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Zu¨rich Swiss Federal Institute of Technology Zurich Structured eigenvalue condition numbers and linearizations for matrix polynomials B. Adhikari∗, R. Alam† and D. Kressner Research Report No. 2009-01 January 2009 Seminar fu¨r Angewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Zu¨rich Switzerland ∗Department of Mathematics, Indian Institute of Technology Guwahati, India, E-mail: [email protected] †Department of Mathematics, Indian Institute of Technology Guwahati, India, E-mail: rafi[email protected], rafi[email protected], Fax: +91-361-2690762/2582649. Structured eigenvalue condition numbers and linearizations for matrix polynomials Bibhas Adhikari∗ Rafikul Alam† Daniel Kressner‡. Abstract. This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix poly- nomials. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Recently, linearizations have been classified for which the pencil reflects the structure of the original polynomial. A question of practical impor- tance is whether this process of linearization increases the sensitivity of the eigenvalue with respect to structured perturbations. For all structures under consideration, we show that this is not the case: there is always a linearization for which the structured condition number of an eigenvalue does not differ significantly. This implies, for example, that a structure-preserving algorithm applied to the linearization fully benefits from a potentially low structured eigenvalue condition number of the original matrix polynomial. Keywords. Eigenvalue problem, matrix polynomial, linearization, structured condition num- ber.
    [Show full text]
  • Condition Number Bounds for Problems with Integer Coefficients*
    Condition Number Bounds for Problems with Integer Coefficients* Gregorio Malajovich Departamento de Matem´atica Aplicada, Universidade Federal do Rio de Janeiro. Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil. E-mail: [email protected] An explicit a priori bound for the condition number associated to each of the following problems is given: general linear equation solving, least squares, non-symmetric eigenvalue problems, solving univariate polynomials, and solv- ing systems of multivariate polynomials. It is assumed that the input has integer coefficients and is not on the degeneracy locus of the respective prob- lem (i.e., the condition number is finite). Our bounds are stated in terms of the dimension and of the bit-size of the input. In the same setting, bounds are given for the speed of convergence of the following iterative algorithms: QR iteration without shift for the symmetric eigenvalue problem, and Graeffe iteration for univariate polynomials. Key Words: condition number, height, complexity 1. INTRODUCTION In most of the numerical analysis literature, complexity and stability of numerical algorithms are usually estimated in terms of the problem instance dimension and of a \condition number". For instance, the complexity of solving an n n linear system Ax = b is usually estimated in terms of the dimension n×(when the input actually consists of n(n + 1) real numbers) and the condition number κ(A) (see section 2.1 for the definition of the condition number). There is a set of problems instances with κ(A) = , and in most cases it makes no sense to solve these instances.
    [Show full text]
  • Inverse Littlewood-Offord Theorems and the Condition Number Of
    Annals of Mathematics, 169 (2009), 595{632 Inverse Littlewood-Offord theorems and the condition number of random discrete matrices By Terence Tao and Van H. Vu* Abstract Consider a random sum η1v1 + ··· + ηnvn, where η1; : : : ; ηn are indepen- dently and identically distributed (i.i.d.) random signs and v1; : : : ; vn are inte- gers. The Littlewood-Offord problem asks to maximize concentration probabil- ities such as P(η1v1+···+ηnvn = 0) subject to various hypotheses on v1; : : : ; vn. In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v1; : : : ; vn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number. 1. Introduction Let v be a multiset (allowing repetitions) of n integers v1; : : : ; vn. Consider a class of discrete random walks Yµ,v on the integers Z, which start at the origin and consist of n steps, where at the ith step one moves backwards or forwards with magnitude vi and probability µ/2, and stays at rest with probability 1−µ. More precisely: Definition 1.1 (Random walks). For any 0 ≤ µ ≤ 1, let ηµ 2 {−1; 0; 1g denote a random variable which equals 0 with probability 1 − µ and ±1 with probability µ/2 each. In particular, η1 is a random sign ±1, while η0 is iden- tically zero.
    [Show full text]
  • Computing the Jordan Structure of an Eigenvalue∗
    SIAM J. MATRIX ANAL.APPL. c 2017 Society for Industrial and Applied Mathematics Vol. 38, No. 3, pp. 949{966 COMPUTING THE JORDAN STRUCTURE OF AN EIGENVALUE∗ NICOLA MASTRONARDIy AND PAUL VAN DOORENz Abstract. In this paper we revisit the problem of finding an orthogonal similarity transformation that puts an n × n matrix A in a block upper-triangular form that reveals its Jordan structure at a particular eigenvalue λ0. The obtained form in fact reveals the dimensions of the null spaces of i (A − λ0I) at that eigenvalue via the sizes of the leading diagonal blocks, and from this the Jordan structure at λ0 is then easily recovered. The method starts from a Hessenberg form that already reveals several properties of the Jordan structure of A. It then updates the Hessenberg form in an efficient way to transform it to a block-triangular form in O(mn2) floating point operations, where m is the total multiplicity of the eigenvalue. The method only uses orthogonal transformations and is backward stable. We illustrate the method with a number of numerical examples. Key words. Jordan structure, staircase form, Hessenberg form AMS subject classifications. 65F15, 65F25 DOI. 10.1137/16M1083098 1. Introduction. Finding the eigenvalues and their corresponding Jordan struc- ture of a matrix A is one of the most studied problems in numerical linear algebra. This structure plays an important role in the solution of explicit differential equations, which can be modeled as n×n (1.1) λx(t) = Ax(t); x(0) = x0;A 2 R ; where λ stands for the differential operator.
    [Show full text]
  • High Probability Analysis of the Condition Number of Sparse Polynomial Systems
    High Probability Analysis of the Condition Number of Sparse Polynomial Systems Gregorio Malajovich 1 Departamento de Matem´atica Aplicada, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil. J. Maurice Rojas 2 Department of Mathematics, Texas A&M University, TAMU 3368, College Station, Texas 77843-3368, USA. Abstract Let f := (f 1,...,f n) be a polynomial system with an arbitrary fixed n-tuple of supports and random coefficients. Our main result is an upper bound on the prob- ability that the condition number of f in a region U is larger than 1/ε. The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying variances) are all identical. We also consider polynomials with real coefficients and give bounds for the ex- pected number of real roots and (restricted) condition number. Using a K¨ahler geometric framework throughout, we also express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form, thus n recovering the classical mixed volume when U = (C∗) . Key words: mixed volume, condition number, polynomial systems, sparse, random. Email addresses: [email protected] (Gregorio Malajovich), [email protected] (J. Maurice Rojas). URLs: http://www.labma.ufrj.br/~gregorio (Gregorio Malajovich), http://www.math.tamu.edu/~rojas (J. Maurice Rojas). 1 Partially supported by CNPq grant 303472/02-2, by CERG (Hong Kong) grants 9040393, 9040402, and 9040188, by FAPERJ and by Fundac˜a¸o Jos´ePel´ucio Ferreira.
    [Show full text]
  • Condition Number and Matrices
    Condition number and matrices Felipe Bottega Diniz March 16, 2017 Abstract It is well known the concept of the condition number κ(A)= A A−1 , where A is k kk k a n n real or complex matrix and the norm used is the spectral norm. Although it × is very common to think in κ(A) as “the” condition number of A, the truth is that condition numbers are associated to problems, not just instance of problems. Our goal is to clarify this difference. We will introduce the general concept of condition number and apply it to the particular case of real or complex matrices. After this, we will introduce the classic condition number κ(A) of a matrix and show some known results. Contents 1 Introduction 1 2 Condition number of matrix problems 3 3 Condition number and ill-posed problems 5 4 Componentwise condition number 6 5 Conclusions 12 1 Introduction arXiv:1703.04547v1 [math.GM] 13 Mar 2017 Let X,Y be normed vector spaces and let f : X Y be a function. We will consider → X as the input space, Y the output space and f the problem. For example, we can have X = Cn×n with any norm, Y = C with absolute value and f(A) = det(A) for all A X. ∈ Therefore, f models the problem of computing the determinant. For any input x X 0 and a perturbationx ˜ X of x, we define the absolute error ofx ˜ with respect to∈x by\{ } ∈ AbsError(˜x, x)= x˜ x k − k and the relative error by x˜ x RelError(˜x, x)= k − k.
    [Show full text]
  • 1 on the Condition of Numerical Problems and the Numbers That
    1 On the Condition of Numerical Problems and the Numbers that Measure It 1.1 The Size of Errors Since none of the numbers we take out from logarithmic or trigonometric tables admit of absolute precision, but are all to a certain extent approximate only, the results of all calculations performed by the aid of these numbers can only be approxi- mately true. [. ] It may happen, that in special cases the effect of the errors of the tables is so augmented that we may be obliged to reject a method, otherwise the best, and substitute another in its place. Carl Friedrich Gauss, Theoria Motus The heroes of numerical mathematics (Euler, Gauss, Lagrange, . ) developed a good number of the algorithmic procedures which make the essence of Numerical Analysis. At the core of these advances there was the invention of calculus. And underlying the latter, the field of real numbers. The dawn of the digital computer, in the decade of the 40s, allowed the execution of these procedures on increasingly large data, an advance which, however, made even more patent the fact that real numbers cannot be encoded with a finite number of bits and, therefore, that computers had to work with approximations only. With the increased length of the computations the sys- tematic rounding of all occurring quantities could now accumulate to a higher extend. Occasionally, as already remarked by Gauss, the errors affecting the outcome of a computation were so big as to make it irrelevant. Expressions like “the error is big” beg the question, how does one measure an error? To approach this question, let us first assume that the object whose error we are considering is a single number x encoding a quantity which may take values on an open real interval.
    [Show full text]
  • Chapter 9 PRECONDITIONING
    Chapter 9 PRECONDITIONING 9.1 Preconditioned Krylov Subspace Methods 9.2 Preconditioned Conjugate Gradient 9.3 Preconditioned Generalized Minimal Residual 9.4 Relaxation Method Preconditioners 9.5 Incomplete LU Factorization Preconditioners 9.6 Approximate Inverse Preconditioners 9.1 Preconditioned Krylov Subspace Methods Saad, Sections 9.0, 9.1 The rate of convergence of a Krylov subspace method for a linear system Ax = b depends on the condition number of the matrix A. Therefore, if we have a matrix M which is a crude approximation to A, M −1A is closer to the identity than is A and should have a smaller condition number, and it would be expected that a Krylov subspace method would converge faster for the “preconditioned” system M −1Ax = M −1b. For example, choosing M to be the diagonal part of A can be quite helpful. This is a useful approach only if computing M −1v for an arbitrary vector v is cheap. Such a matrix M is called a preconditioner or, more precisely, a left preconditioner. In the case of a right preconditioner,onesolves AM −1u = b where x = M −1u. 175 And if M is in factored form M = MLMR,onecanusethisasasplit preconditioner by solving −1 −1 −1 −1 ML AMR u = ML b where x = MR u. −1 −1 −1 −1 Note that M A, AM and ML AMR are similar matrices, so they have the same eigen- values. For convenience write the given system as A(x − x0)=r0 where r0 = b − Ax0, so that x − x0 is the unknown, and consider a split preconditioner M = MLMR.Aleft preconditioner is the case MR = I and a right preconditioner is the case ML = I.
    [Show full text]
  • CPSC 303: WHAT the CONDITION NUMBER DOES and DOES NOT TELL US Contents 1. Motivating Examples from Interpolation 2 1.1. an N
    CPSC 303: WHAT THE CONDITION NUMBER DOES AND DOES NOT TELL US JOEL FRIEDMAN Contents 1. Motivating Examples from Interpolation2 1.1. An n = 1 Example: the Tangent Line of Calculus2 1.2. Degeneracy in Interpolation is Not Degenerate and Yields Derivatives3 1.3. An n = 2 Example With One Derivative3 1.4. A Doubly Degenate Example and Taylor Series3 2. More Motivation (From General Linear Algebra) for the Condition Number5 2.1. Sensitivity in General n × n Systems5 2.2. Sensitivity in General Interpolation5 2.3. Optimal Sensitivity: Diagonal Systems6 2.4. Cancellation6 2.5. A Mixed Signs Example6 3. Cancellation, Awkwardness and Norms, and Relative Error7 3.1. Common Norms and Awkward Norms7 3.2. The Lp-Norm (or `p-Norm or p-Norm)8 3.3. Cancellation and the Triangle Inequality8 3.4. Relative Error9 3.5. Further Remarks on Norms9 4. The L2-Norm (or 2-Norm) of a Matrix 10 5. The Lp-Norm (or p-Norm) of a Matrix 12 6. Some 2 × 2 Matrix Norm Formulas 13 7. The (L2- and) Lp-Condition Number of a Matrix 14 8. Interpolation: What The Condition Number Does and Does Not Tell Us 15 8.1. Formulas for the Inverse 15 8.2. The 2 × 2 Interpolation for a Tangent Line 16 8.3. A 3 × 3, Doubly Degenerate Example 17 9. The Doubly-Normed Condition Number 18 Exercises 19 Copyright: Copyright Joel Friedman 2020. Not to be copied, used, or revised without explicit written permission from the copyright owner. Research supported in part by an NSERC grant.
    [Show full text]
  • Structured Eigenvalue Condition Numbers
    Structured Eigenvalue Condition Numbers Karow, Michael and Kressner, Daniel and Tisseur, Françoise 2006 MIMS EPrint: 2006.45 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 STRUCTURED EIGENVALUE CONDITION NUMBERS MICHAEL KAROW∗, DANIEL KRESSNER∗, AND FRANC¸OISE TISSEUR† Abstract. This paper investigates the effect of structure-preserving perturbations on the eigen- values of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo symmetric, persymmetric, skew- symmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number. Key words. Structured eigenvalue problem, condition number, Jordan algebra, Lie algebra, automorphism group, symplectic,
    [Show full text]
  • 7.4 Matrix Norms and Condition Numbers This Section Considers the Accuracy of Computed Solutions to Linear Systems
    Ch 7: Numerical Linear Algebra 7.4 Matrix Norms and Condition Numbers This section considers the accuracy of computed solutions to linear systems. At every step, the computer will roundoff, and we would like to know how far the computer answer is from the real answer. 1. Let A be an m × n. Then (a) recall the Frobenius norm of an qPn Pm 2 jjAjjF = i=1 j=1 aij = p 2 2 2 2 2 2 2 2 a11 + a12 + a13 + ::: + a1n + a21 + a22 + ::: + a2n + ::: + amn (b) the 1-norm: m X jjAjj1 = max jaijj = 1≤j≤n i=1 = max (ja11j + ja21j + ja31j + ::: + jam1j); (ja12j + ja22j + ::: + jam2j);:::; (ja1nj + ja2nj + ::: + jamnj); i.e. it is the maximum among the sums of the absolute values of the elements of each column (c) the 1-norm: n X jjAjj1 = max jaijj = 1≤i≤m j=1 = max (ja11j + ja12j + ja13j + ::: + ja1nj); (ja21j + ja22j + ::: + ja2nj);:::; (jam1j + jam2j + ::: + jamnj); i.e. it is the maximum among the sums of the absolute values of the elements of each row (d) the 2-norm: jjAxjj2 jjAjj2 = max ; x6=0 jjxjj2 p 2 2 2 where jjxjj2 = x1 + x2 + : : : xn and similarly for jjAxjj2. or jjAjj2 = σ1; T where σ1 is the largest singular value in the SVD of A = UΣV . As a consequence −1 1 jjA jj2 = ; σn T where σn is the smallest nonzero singular value in the SVD of A = UΣV . 2. some properties of the Frobenius norm: q Pn 2 (a) jjAjjF = i=1 jjajjj2, i.e. it jjAjjF is the square root of the sums of the squares of the standard 2-norms of its columns (similarly it could be computed using rows instead of columns) 1 (b) jjAxjj2 ≤ jjAjjF jjxjj2 (c) jjABjjF ≤ jjAjjF jjBjjF n n (d) jjA jjF ≤ jjAjjF (this is a consequence of the property above) 3.
    [Show full text]
  • Lecture 2 Linear Algebra Review Condition Numbers
    18.409 The Behavior of Algorithms in Practice 2/12/2 Lecture 2 Lecturer: Dan Spielman Scribe: Steve Weis Linear Algebra Review A n x n matrix has n singular values. For a matrix A, the largest singular value is denoted as σn(A). Similarly, the smallest is denoted as σ1(A). They are defined as follows: ||Ax|| σn(A) = ||A|| = max x ||x|| −1 −1 ||Ax|| σ1(A) = ||A || = min x ||x|| There are several other equivalent definitions: q q T T {σn(A), . , σ1(A)} = { λn(A A),..., λ1(A A)} ||Ax|| ||Ax|| σi(A) = min max = max min subspacesS,dim(S)=i x∈S ||x|| subspacesS,dim(S)=(n−i+1) x∈S ||x|| Another classic definition is to take a unit sphere and apply A to it, resulting in some hyper-ellipse. σn will be the length of the largest axis, σn−1 will be the length of the next largest orthogonal axis, etc.. Exercise: Prove that every real matrix A has a singular-value decompsition as A = USV , where U and V are orthogonal matrices and S is non-negative diagonal, and all entries in U, S, and V are real. Condition Numbers The singular values define a condition number of a matrix as follows: σn(A) ||A|| κ(A) := = −1 −1 σ1(A) ||A || Lemma 1. If Ax = b and A(x + δx) = b + δb then ||δx|| ||δb|| ≤ κ(A) ||x|| ||b|| 1 Proof of Lemma 1: ||δb|| Aδx = δb ⇒ δx = A−1δb ⇒ ||b|| ≤ ||δb|| · ||A−1|| = σ1(A) 1 σ (A) Ax = b ⇒ ||b|| ≤ ||A|| · ||x|| = σ (A) · ||x|| ⇒ ≤ n n ||x|| ||b|| Lemma 1 follows from these two inequalities.
    [Show full text]