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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 017 902 888
Highham, Nicolas J.; Tisseur, Francoise:
More on pseudospectra for polynomial eigenvalue problems and applications in control theory
Linear Algebra Appl. 351-352, 435 - 453 (2002).
15A22Matrix pencils
Primary Classification:
65F15Eigenvalues, eigenvectors
Secondary Classification:
47A55Perturbation theory
polynomial eigenvalue problem; pseudospectrum; nearest nonregular polynomial; nearest uncontrollable dynamical system; structured perturbations

Next to the usual linear spectral (or eigenvalue) problem A x = e x
one finds the generalized one, A x = e B x. Moving on, a
square-matrix polynomial problem follows, P(e) x = 0. Its non-square
form appeals finally to the authors motivated (say, by the theory of
games) to study the ``infinite" eigenvalues e = a/b with b = 0. They
contemplate the homogeneous polynomial and add
perturbations, [P(a,b)+ D P'(a,b)] x = 0. In this way the authors
define the (complex) pseudospectrum [= a set of pairs
(a,b)] and show how it can be calculated via minimization of P x over
x with unit norm (cf. their main Theorem 2.1). In the sequel they
illustrate the concept (on the Wilkinson's pathological example),
suggest a vizualization on the Riemann sphere (a projection which
treats also the information about the infinite eigenvalues properly)
and outline a few applications [studying the distance from the
nearest nonregular matrix polynomial or from a domain of
uncontrollability (= impossibility to reach some final states) in a
dynamical system]. Finally they extend the concept, in the spirit of
their preceding paper [21], to the so called structured pseudospectra
and illustrate their merits on a damped mass-spring example of the
Tisseur's older paper [20].
Remarks to the editors:
I do not mind replacements of my plain-text equations by their improved TEX forms (then, you could also replace e by lambda, a by \alpha, b by \beta and D by capital \Delta).

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