The irreducible control property in matrix groups

Jan Draisma

doi:10.1016/j.laa.2021.10.020

The irreducible control property in matrix groups

Jan Draisma (Corresponding author)

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Abstract

This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group.

Original language	English
Pages (from-to)	15-29
Number of pages	15
Journal	Linear Algebra and Its Applications
Volume	634
DOIs	https://doi.org/10.1016/j.laa.2021.10.020
Publication status	Published - 1 Feb 2022

Bibliographical note

Funding Information:
The author is partially supported by Vici grant 639.033.514 from the Netherlands Organisation for Scientific Research and project grant 200021_191981 from the Swiss National Science Foundation.

Funding

The author is partially supported by Vici grant 639.033.514 from the Netherlands Organisation for Scientific Research and project grant 200021_191981 from the Swiss National Science Foundation.

Keywords

Matrix factorisations
Matrix groups

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10.1016/j.laa.2021.10.020

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abstract = "This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group.",

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N2 - This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group.

AB - This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group.

KW - Matrix factorisations

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

The irreducible control property in matrix groups

Abstract

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