The change of the Jordan structure of a mtrix under small perturbations
TL;DR: In this paper, a complete description including multiplicity is given for the Jordan structure of a matrix which is a small perturbation of a known Jordan structure, and the problem solved here was solved independently and the other solution has been published in English.
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-10-01 and is currently open access. It has received 43 citations till now.
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TL;DR: In this article, the existence of an n × n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed is solved in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariants.
112 citations
TL;DR: It is proposed that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm.
Abstract: Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem We propose that knowledge of the closure relations, ie, the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm This paper is a continuation of our Part I paper on versal deformations, but it may also be read independently
93 citations
Cites result from "The change of the Jordan structure ..."
...Our results for the closure decision problem are also derived in [ 36 , 17]....
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TL;DR: In this article, the Hasse diagrams G2 and G3 were constructed for the closure ordering on the sets of congruence classes of 2 × 2 and 3 × 3 complex matrices.
32 citations
TL;DR: In this article, the variation of controllability indices and the Jordan structure of a pair of matrices under small perturbations was studied, and it was shown that the Jordan structures of the two matrices A and B can be obtained from the controllable indices.
31 citations
TL;DR: In this article, it was shown that any matrix polynomial in an algebraically closed field can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors.
26 citations
Cites background from "The change of the Jordan structure ..."
...An explicit proof can be found in [6] (see also [4, p....
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Book•
15 Aug 2017
TL;DR: In this paper, a treatment of the classical Riesz-Schauder theory is presented, which takes advantage of the most recent developments in functional analysis. But this treatment is restricted to the case of bounded linear operators on a Hilbert space.
Abstract: Since the appearance of Banach algebra theory, the interaction between the theories ofBanach algebras with involution and that of bounded linear operators on a Hilbert space hasbeen extensively developed The connections of Banach algebras with the theory ofbounded linear operators on a Hilbert space have also evolved, and Calkin Algebras andAlgebras of Operators on Banach Spaces provides an introduction to this set of ideasThe book begins with a treatment of the classical Riesz-Schauder theory which takesadvantage of the most recent developments-some of this material appears here for the firsttime Although the reader should be familiar with the basics of functional analysis, anintroductory chapter on Banach algebras has been included Other topics dealt with includeFredholm operators, semi-Fredholm operators, Riesz operators and Calkin algebrasThis volume will be of direct interest to both graduate students and research mathematicians
221 citations
TL;DR: In this paper, it was shown that given a holomorphic function A on a bounded domain Ω into a space of bounded linear operators between two Banach spaces, it is possible to extend the operators A(λ) by an identity operator IZ in such a way that the extended operator function A(·) ⊕ IZ is equivalent on Ω to a linear function of λ, T − λI.
78 citations
51 citations