Derivatives of Eigenvalues and Eigenvectors of Matrix Functions

Andrew, Alan L.; Chu, Kaiwei; Lancaster, Peter

doi:10.1137/0614061

Cited by 144 publications

(121 citation statements)

References 25 publications

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“…Let ðg v Þ j denote the jth entry of vector g v and k Á k 1 denote the L 1 norm or the sum norm. Section 12 in Andrew et al (1993) shows that …”

Section: Resultsmentioning

confidence: 99%

“…Although the explicit expression of these functions for a general k is expected to be complicated, a general result in Andrew et al (1993) shows that these functions are in fact analytic. Moreover, it is not necessary to use these explicit expressions in the computation.…”

Section: Article In Pressmentioning

confidence: 99%

“…In this section, let $ðyÞ ¼ y in Assumptions 2.3 and 2.5. The identification results imply that one can express the unknown density f x Ã jzw as follows: Although the expression of the function c is complicated, a general result in Andrew et al (1993) shows that the function c is a well-behaved analytic function around g 0 . Therefore,…”

Section: Article In Pressmentioning

confidence: 99%

See 2 more Smart Citations

Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution

2008

Journal of Econometrics

218

235

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This paper provides a general solution to the problem of identification and estimation of nonlinear models with misclassification error in a general discrete explanatory variable using instrumental variables. The misclassification error is allowed to be correlated with all the explanatory variables in the model. It is not enough to identify the model by simply generalizing the identification in the binary case with a claim that the number of restrictions is no less than that of the unknowns. Such a claim requires solving a complicated nonlinear system of equations. This paper introduces a matrix diagonalization technique which allows one to easily find the unique solution of the system. The solution shows that the latent model can be expressed as an explicit function of directly observed distribution functions. Therefore, the latent model is nonparametrically identifiable and directly estimable using instrumental variables. The results show that certain monotonicity restrictions on the latent model may lead to its identification with virtually no restrictions on the misclassification probabilities. An alternative identification condition suggests that the nonparametric identification may rely on the belief that people always have a higher probability of telling the truth than of misreporting. The nonparametric identification in this paper directly leads to a ffiffi ffi n p -consistent semiparametric estimator. The Monte Carlo simulation and empirical illustration show that the estimator performs well with a finite sample and real data. r 2007 Elsevier B.V. All rights reserved.JEL classification: C14; C41

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“…Let ðg v Þ j denote the jth entry of vector g v and k Á k 1 denote the L 1 norm or the sum norm. Section 12 in Andrew et al (1993) shows that …”

Section: Resultsmentioning

confidence: 99%

Section: Article In Pressmentioning

confidence: 99%

Section: Article In Pressmentioning

confidence: 99%

See 1 more Smart Citation

Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution

2008

Journal of Econometrics

218

235

View full text Add to dashboard Cite

show abstract

“…If y and x are the left and the right eigenvector of a simple eigenvalue λ 0 of a nonlinear eigenvalue problem T (λ)x = 0, where T is differentiable, then it is wellknown that y * T (λ)x = 0, see, e.g., [1,14]. The following proposition generalizes this relation to the nonlinear two-parameter eigenvalue problem.…”

Section: Theorem 23 ([12 Theorem 4]) a Basis For The Kernel Of ∆ =mentioning

confidence: 88%

On the singular two-parameter eigenvalue problem

Muhič¹,

Plestenjak²

2009

ELA

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Abstract. In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above relation to singular two-parameter eigenvalue problems is extended, and it is shown that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the associated system of generalized eigenvalue problems agree. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems.

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“…It comes from applying the Galerkin method to a PDE describing the wave motion of a vibrating string with clamped ends in a spatially inhomogeneous environment [5], [7]. The quadratic Q is elliptic; the eigenvalues are nonreal and have absolute values in the interval [1,25]. Figure 8.1 shows the condition numbers κ L (α, β) for the DL(Q) linearization with v = e 1 and the first companion linearization, the condition number κ P (α, β) for Q, and the angular errors in the eigenvalues computed by applying the QZ algorithm to the two linearizations.…”

Section: Numerical Experimentsmentioning

confidence: 99%

The Conditioning of Linearizations of Matrix Polynomials

Higham

Mackey²,

Tisseur

2006

SIAM J. Matrix Anal. & Appl.

105

160

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Abstract. The standard way of solving the polynomial eigenvalue problem of degree m in n × n matrices is to "linearize" to a pencil in mn × mn matrices and solve the generalized eigenvalue problem. For a given polynomial, P , infinitely many linearizations exist and they can have widely varying eigenvalue condition numbers. We investigate the conditioning of linearizations from a vector space DL(P ) of pencils recently identified and studied by Mackey, Mackey, Mehl, and Mehrmann. We look for the best conditioned linearization and compare the conditioning with that of the original polynomial. Two particular pencils are shown always to be almost optimal over linearizations in DL(P ) for eigenvalues of modulus greater than or less than 1, respectively, provided that the problem is not too badly scaled and that the pencils are linearizations. Moreover, under this scaling assumption, these pencils are shown to be about as well conditioned as the original polynomial. For quadratic eigenvalue problems that are not too heavily damped, a simple scaling is shown to convert the problem to one that is well scaled. We also analyze the eigenvalue conditioning of the widely used first and second companion linearizations. The conditioning of the first companion linearization relative to that of P is shown to depend on the coefficient matrix norms, the eigenvalue, and the left eigenvectors of the linearization and of P . The companion form is found to be potentially much more ill conditioned than P , but if the 2-norms of the coefficient matrices are all approximately 1 then the companion form and P are guaranteed to have similar condition numbers. Analogous results hold for the second companion form. Our results are phrased in terms of both the standard relative condition number and the condition number of Dedieu and Tisseur for the problem in homogeneous form, this latter condition number having the advantage of applying to zero and infinite eigenvalues.

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Derivatives of Eigenvalues and Eigenvectors of Matrix Functions

Cited by 144 publications

References 25 publications

Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution

Identification and estimation of nonlinear models with misclassification error using instrumental variables: A general solution

On the singular two-parameter eigenvalue problem

The Conditioning of Linearizations of Matrix Polynomials

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