The algebraic structure of pencils and block Toeplitz matrices
type: Item
instance of: scholarly article
title: The algebraic structure of pencils and block Toeplitz matrices
zbMATH Open document ID: 0936.15010
DOI: 10.1016/S0024-3795(98)00022-6
author: Daniel Boley
published in: Linear Algebra and its Applications
publication date: 2000-05-18T00:00:00Z
review text: The paper restates several results on the Kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation, and states some new results on the Kronecker structure after a new row is appended or a column deleted. The results are simple consequences of a theory of majorization for semi-infinite integer sequences. The bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time-invariant dynamical system. The known relations between the Kronecker indices, the Segre characteristics, and the nullities of certain block Toeplitz matrices constructed from the original pencils are illustrated. How those nullities change when the pencil is affected by a perturbation or a new row is also discussed.
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B25
zbMATH DE Number: 1355352
zbMATH Keywords: matrix pencil
zbMATH Keywords: perturbation
zbMATH Keywords: Kronecker structure
zbMATH Keywords: majorization for semi-infinite integer sequences
zbMATH Keywords: controllability indices
zbMATH Keywords: linear time-invariant dynamical system
zbMATH Keywords: Kronecker indices
zbMATH Keywords: Segre characteristics
zbMATH Keywords: block Toeplitz matrices
reviewed by: Rodica Covaci
MaRDI profile type: Publication
cites work: Placing zeroes and the Kronecker canonical form
cites work: Points of Continuity of the Kronecker Canonical Form
cites work: Young diagrams and determinantal varieties
cites work: The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part II
cites work: Semi-stability of sums of partial multiplicities under additive perturbation
cites work: A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils. Part II: A Stratification-Enhanced Staircase Algorithm
cites work: A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils. Part I: Versal Deformations
cites work: The Set of 2-by-3 Matrix Pencils — Kronecker Structures and Their Transitions under Perturbations
cites work: Computation of zeros of linear multivariable systems
cites work: Q3254327
cites work: Perturbation of linear control systems
cites work: Orbit closures of singular matrix pencils
cites work: Q3902876
cites work: On the Segré, Weyr characteristics of right (left) regular matrix pencils
cites work: Right, left characteristic sequences and column, row minimal indices of a singular pencil
cites work: Q3947818
cites work: The change of the Jordan structure of a matrix under small perturbations
cites work: Inequalities: theory of majorization and its applications
cites work: On perturbations and the equivalence orbit of a matrix pencil
cites work: Q3274173
cites work: The computation of Kronecker's canonical form of a singular pencil
cites work: The generalized eigenstructure problem in linear system theory
cites work: On the determination of the Smith-Macmillan form of a rational matrix from its Laurent expansion
description: scientific article; zbMATH DE number 1355352