Pairs of maps preserving singularity on subsets of matrix algebras

A. E. Guterman; A. M. Maksaev; V. V. Promyslov

doi:10.1016/j.laa.2022.02.035

Pairs of maps preserving singularity on subsets of matrix algebras

A. E. Guterman, A. M. Maksaev, V. V. Promyslov

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

3 Scopus citations

Abstract

Let F be an algebraically closed field and M_n be the n×n matrix algebra over F. A total graph of the full matrix algebra is the graph with M_n as vertices, and two distinct matrices A,B are adjacent if and only if A+B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of M_n preserving the singularity of matrix pencils A+λB. In particular, we characterize maps T₁,T₂:M_n→M_n satisfying the condition A+λB is singular if and only if T₁(A)+λT₂(B) is singular, for any A,B∈M_n and any non-zero λ∈F. Namely, we prove that in this case T₁=T₂ and they are of the form T₁(A)=T₂(A)=PAQ for all A∈M_n, or of the form T₁(A)=T₂(A)=PA^tQ for all A∈M_n, where P,Q∈M_n are non-singular matrices.

Original language	English
Pages (from-to)	1-27
Number of pages	27
Journal	Linear Algebra and Its Applications
Volume	644
DOIs	https://doi.org/10.1016/j.laa.2022.02.035
State	Published - 1 Jul 2022
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Funding

The work of the second author is financially supported by a scholarship of Theoretical Physics and Mathematics Advancement Foundation “ BASIS ” (grant No 21-8-3-18-1 ). The authors are grateful to Bojan Kuzma whose question inspired us to generalize our previous statements and proofs to a pair of maps, which led the main results of the paper. We also thank the anonymous referee for useful comments. The work of the second author is financially supported by a scholarship of Theoretical Physics and Mathematics Advancement Foundation “BASIS” (grant No 21-8-3-18-1).

Funders	Funder number
Bojan Kuzma
Foundation for the Advancement of Theoretical Physics and Mathematics	21-8-3-18-1

Keywords

Automorphisms of graphs
Determinant
Matrix pencils
Preserver

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10.1016/j.laa.2022.02.035

Cite this

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title = "Pairs of maps preserving singularity on subsets of matrix algebras",

abstract = "Let F be an algebraically closed field and Mn be the n×n matrix algebra over F. A total graph of the full matrix algebra is the graph with Mn as vertices, and two distinct matrices A,B are adjacent if and only if A+B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of Mn preserving the singularity of matrix pencils A+λB. In particular, we characterize maps T1,T2:Mn→Mn satisfying the condition A+λB is singular if and only if T1(A)+λT2(B) is singular, for any A,B∈Mn and any non-zero λ∈F. Namely, we prove that in this case T1=T2 and they are of the form T1(A)=T2(A)=PAQ for all A∈Mn, or of the form T1(A)=T2(A)=PAtQ for all A∈Mn, where P,Q∈Mn are non-singular matrices.",

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doi = "10.1016/j.laa.2022.02.035",

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T1 - Pairs of maps preserving singularity on subsets of matrix algebras

AU - Guterman, A. E.

AU - Maksaev, A. M.

AU - Promyslov, V. V.

PY - 2022/7/1

Y1 - 2022/7/1

N2 - Let F be an algebraically closed field and Mn be the n×n matrix algebra over F. A total graph of the full matrix algebra is the graph with Mn as vertices, and two distinct matrices A,B are adjacent if and only if A+B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of Mn preserving the singularity of matrix pencils A+λB. In particular, we characterize maps T1,T2:Mn→Mn satisfying the condition A+λB is singular if and only if T1(A)+λT2(B) is singular, for any A,B∈Mn and any non-zero λ∈F. Namely, we prove that in this case T1=T2 and they are of the form T1(A)=T2(A)=PAQ for all A∈Mn, or of the form T1(A)=T2(A)=PAtQ for all A∈Mn, where P,Q∈Mn are non-singular matrices.

AB - Let F be an algebraically closed field and Mn be the n×n matrix algebra over F. A total graph of the full matrix algebra is the graph with Mn as vertices, and two distinct matrices A,B are adjacent if and only if A+B is singular. The characterization of all the automorphisms of the total graph is an open question. Motivated by this problem, we study pairs of maps on a subset of Mn preserving the singularity of matrix pencils A+λB. In particular, we characterize maps T1,T2:Mn→Mn satisfying the condition A+λB is singular if and only if T1(A)+λT2(B) is singular, for any A,B∈Mn and any non-zero λ∈F. Namely, we prove that in this case T1=T2 and they are of the form T1(A)=T2(A)=PAQ for all A∈Mn, or of the form T1(A)=T2(A)=PAtQ for all A∈Mn, where P,Q∈Mn are non-singular matrices.

KW - Automorphisms of graphs

KW - Determinant

KW - Matrix pencils

KW - Preserver

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JO - Linear Algebra and Its Applications

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ER -

Pairs of maps preserving singularity on subsets of matrix algebras

Abstract

Bibliographical note

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Keywords

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