Pairs of maps preserving singularity on subsets of matrix algebras

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

doi:https://doi.org/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

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

Keywords

Automorphisms of graphs
Determinant
Matrix pencils
Preserver

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.2022.02.035

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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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author = "Guterman, {A. E.} and Maksaev, {A. M.} and Promyslov, {V. V.}",

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doi = "https://doi.org/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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U2 - https://doi.org/10.1016/j.laa.2022.02.035

DO - https://doi.org/10.1016/j.laa.2022.02.035

M3 - مقالة

SN - 0024-3795

VL - 644

SP - 1

EP - 27

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Pairs of maps preserving singularity on subsets of matrix algebras

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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