On condition numbers of a nondefective multiple eigenvalue

Summary This paper concerns with measures of the sensitivity of a nondefective multiple eigenvalue of a matrix. Different condition numbers are introduced starting from directional derivatives of the multiple eigenvalue. Properties of the condition numbers defined by Stewart and Zhang [4] are studied; especially, the Wilkinson's theorem on matrices with a very ill-conditioned eigenproblem is extended.This research was supported by the Institute for Advanced Computer Studies of the University of Maryland and the Swedish Natural Science Research Council     

On worst-case condition numbers of a nondefective multiple eigenvalue

Summary. This paper is a continuation of the author [6] in Numerische Mathematik. Let be a nondefective multiple eigenvalue of multiplicity of an complex matrix , and let be the secants of the canonical angles between the left and right invariant subspaces of corresponding to the multiple eigenvalue . The analysis of this paper shows that the quantities are the worst-case condition numbers of the multiple eigenvalue . Received September 28, 1992 / Revised version received January 18, 1994     

Eigenvalues of graded matrices and the condition numbers of a multiple eigenvalue

Summary This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of the canonical angles between the left and right invariant of a multiple eigenvalue tend to characterize its behavior when its matrix is slightly perturbed.This work was supported in part by the Air Force Office of Sponsored Research under Contract AFOSR-87-0188     

On the condition numbers of a multiple eigenvalue of a generalized eigenvalue problem

For standard eigenvalue problems, closed-form expressions for the condition numbers of a multiple eigenvalue are known. In particular, they are uniformly 1 in the Hermitian case and generally take different values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers. Our main result is that a multiple eigenvalue generally has multiple condition numbers, even in the Hermitian definite case. The condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors.     

Structured eigenvalue condition numbers and linearizations for matrix polynomials

This work is concerned with eigenvalue problems for structured matrix polynomials, including complex symmetric, Hermitian, even, odd, palindromic, and anti-palindromic matrix polynomials. 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 importance is whether this process of linearization significantly increases the eigenvalue sensitivity with respect to structured perturbations. For all structures under consideration, we show that this cannot happen if the matrix polynomial is well scaled: there is always a structured linearization for which the structured eigenvalue condition number does not differ much. 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.     

On condition numbers of polynomial eigenvalue problems

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented.     

A deflation method for regular matrix pencils

A generalization of the concept of eigenvalue is introduced for a matrix pencil and it is called eigenpencil; an eigenpencil is a pencil itself and it contains part of the spectral information of the matrix pencil. A Wielandt type deflation procedure for regular matrix pencils is developed, using eigenpencils and supposing that they can have both finite and infinite eigenvalues. A numerical example illustrates the proposed method.     

SENSITIVITY OF THE EIGENVALUES OF A DEFECTIVE MATRIX

This paper is concerned with the sensitivity of the eigenvalues of a defective matrixunder small perturbations.The given estimate generalizes all special results of Wilkinson,Stewart,Bauer and Fike,when the eigenvalue is simple and when the matrix is nondefective,and interpretes the phenomenon indicated by Golub and Wilkinson for Multiple eigenvalues     

The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices

A new algorithm for the computation of eigenvalues of a nonsymmetric matrix pencil is described. It is a generalization of the shifted and inverted Lanczos (or Arnoldi) algorithm, in which several shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil, gives Ritz approximations to the solution of the original pencil. It is shown how complex shifts can be used to compute a real block Hessenberg pencil to a real matrix pair.Two applicationx, one coming from an aircraft stability problem and the other from a hydrodynamic bifurcation, have been tested and results are reported.Dedicated to Carl-Erik Fröberg on the occasion of his 75th birthday.     

On close eigenvalues of tridiagonal matrices

Summary. A symmetric tridiagonal matrix with a multiple eigenvalue must have a zero subdiagonal element and must be a direct sum of two complementary blocks, both of which have the eigenvalue. Yet it is well known that a small spectral gap does not necessarily imply that some is small, as is demonstrated by the Wilkinson matrix. In this note, it is shown that a pair of close eigenvalues can only arise from two complementary blocks on the diagonal, in spite of the fact that the coupling the two blocks may not be small. In particular, some explanatory bounds are derived and a connection to the Lanczos algorithm is observed. The nonsymmetric problem is also included. Received April 8, 1992 / Revised version received September 21, 1994     

Variational Analysis of the Spectral Abscissa at a Matrix with a Nongeneric Multiple Eigenvalue

The spectral abscissa is a fundamental map from the set of complex matrices to the real numbers. Denoted α and defined as the maximum of the real parts of the eigenvalues of a matrix X, it has many applications in stability analysis of dynamical systems. The function α is nonconvex and is non-Lipschitz near matrices with multiple eigenvalues. Variational analysis of this function was presented in Burke and Overton (Math Program 90:317–352, 2001), including a complete characterization of its regular subgradients and necessary conditions which must be satisfied by all its subgradients. A complete characterization of all subgradients of α at a matrix X was also given for the case that all active eigenvalues of X (those whose real part equals α(X)) are nonderogatory (their geometric multiplicity is one) and also for the case that they are all nondefective (their geometric multiplicity equals their algebraic multiplicity). However, necessary and sufficient conditions for all subgradients in all cases remain unknown. In this paper we present necessary and sufficient conditions for the simplest example of a matrix X with a derogatory, defective multiple eigenvalue.     

Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint

We consider minimization of a quadratic objective function subject to a sign-indefinite quadratic equality constraint. We derive necessary and sufficient conditions for the existence of solutions to the constrained minimization problem. These conditions involve a generalized eigenvalue of the matrix pencil consisting of a symmetric positive-semidefinite matrix and a symmetric indefinite matrix. A complete characterization of the solution set to the constrained minimization problem in terms of the eigenspace of the matrix pencil is provided.     

Construction of a canonic basis for matrices and pencils of matrices

An algorithm is offered, which with insignificant modifications permits; 1) the finding of a canonic basis of the root sub space corresponding to a prescribed eigenvalue of a matrix; 2) the finding of chains of associated vectors to the eigenvectors corresponding to a prescribed eigenvalue of a regular linear pencil; 3) the finding of chains of generalized associated vectors corresponding to a prescribed eigenvalue of a regular kernel of a singular linear pencil of complete column rank of two matrices; 4) the finding of linearly independent polynomial solutions of a singular linear pencil. The algorithm consists in the construction of a finite sequence of certain auxiliary matrices the choice of which depends on the problem being solved and in the construction of a sequence of their null-spaces, enabling the obtaining of all necessary information on the unknown vectors of the canonic basis of the problem being solved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 90, pp. 46–62, 1979.     

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Summary. We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix , or a pair of left and right deflating subspaces of a regular matrix pencil . This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm. Received September 20, 1994 / Revised version received February 5, 1996     

Calculating the minimal/maximal eigenvalue of symmetric parameterized matrices using projection

In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters ω and we are interested in the minimal eigenvalue of a matrix pencil ( A , B ) with A , B symmetric and B positive definite. If ω can be interpreted as the realization of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of ω . Because this is costly for large matrices, we are looking for a small parameterized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible nonsmoothness of the minimal eigenvalue. The small‐scale eigenvalue problem is obtained by projection of the large‐scale problem. Our main contribution is that, for constructing the subspace, we use multiple eigenvectors and derivatives of eigenvectors. We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.     

Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrixX that satisfies the equationXA =A ^tX, wheret indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.     

Sensitivity of eigenvalues of an unsymmetric tridiagonal matrix

Several relative eigenvalue condition numbers that exploit tridiagonal form are derived. Some of them use triangular factorizations instead of the matrix entries and so they shed light on when eigenvalues are less sensitive to perturbations of factored forms than to perturbations of the matrix entries. A novel empirical condition number is used to show when perturbations are so large that the eigenvalue response is not linear. Some interesting examples are examined in detail.     

Rational Krylov sequence methods for eigenvalue computation

Algorithms to solve large sparse eigenvalue problems are considered. A new class of algorithms which is based on rational functions of the matrix is described. The Lanczos method, the Arnoldi method, the spectral transformation Lanczos method, and Rayleigh quotient iteration all are special cases, but there are also new algorithms which correspond to rational functions with several poles. In the simplest case a basis of a rational Krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of Hessenberg matrices.     

Constraint preconditioning for nonsymmetric indefinite linear systems

This paper introduces and presents theoretical analyses of constraint preconditioning via a Schilders'‐like factorization for nonsymmetric saddle‐point problems. We extend the Schilders' factorization of a constraint preconditioner to a nonsymmetric matrix by using a different factorization. The eigenvalue and eigenvector distributions of the preconditioned matrix are determined. The choices of the parameter matrices in the extended Schilders' factorization and the implementation of the preconditioning step are discussed. An upper bound on the degree of the minimum polynomial for the preconditioned matrix and the dimension of the corresponding Krylov subspace are determined, as well as the convergence behavior of a Krylov subspace method such as GMRES. Numerical experiments are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

对称不定系统的一类预处理子的注记

由于对称正定系统已有很多有效的求解方法,因此将对称的、或者非对称的不定系统转化为对称正定系统就成为解决这类问题的方法之一构造了一类简洁有效的预处理子,将对称不定系统转化为对称正定型,研究了所得预处理系统的谱性质,估计了其谱条件数,推广了现有结论.