Lengths of quasi-commutative pairs of matrices

A. E. Guterman; O. V. Markova; V. Mehrmann

doi:https://doi.org/10.1016/j.laa.2015.11.028

Lengths of quasi-commutative pairs of matrices

A. E. Guterman, O. V. Markova, V. Mehrmann

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.

Original language	English
Pages (from-to)	450-470
Number of pages	21
Journal	Linear Algebra and Its Applications
Volume	498
DOIs	https://doi.org/10.1016/j.laa.2015.11.028
State	Published - 1 Jun 2016
Externally published	Yes

Keywords

Finite-dimensional algebras
Lengths of sets and algebras
Quasi-commuting matrices

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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https://doi.org/10.1016/j.laa.2015.11.028

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title = "Lengths of quasi-commutative pairs of matrices",

abstract = "In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.",

keywords = "Finite-dimensional algebras, Lengths of sets and algebras, Quasi-commuting matrices",

author = "Guterman, {A. E.} and Markova, {O. V.} and V. Mehrmann",

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doi = "https://doi.org/10.1016/j.laa.2015.11.028",

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T1 - Lengths of quasi-commutative pairs of matrices

AU - Guterman, A. E.

AU - Markova, O. V.

AU - Mehrmann, V.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.

AB - In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.

KW - Finite-dimensional algebras

KW - Lengths of sets and algebras

KW - Quasi-commuting matrices

UR - http://www.scopus.com/inward/record.url?scp=84962423471&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.laa.2015.11.028

DO - https://doi.org/10.1016/j.laa.2015.11.028

M3 - مقالة

SN - 0024-3795

VL - 498

SP - 450

EP - 470

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Lengths of quasi-commutative pairs of matrices

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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