Pairs of maps preserving singularity on subsets of matrix algebras

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...

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Bibliographic Details
Published in:Linear algebra and its applications 2022-07, Vol.644, p.1-27
Main Authors: Guterman, A.E., Maksaev, A.M., Promyslov, V.V.
Format: Article
Language:English
Subjects:
Apexes
Automorphisms
Automorphisms of graphs
Determinant
Fields (mathematics)
Graph theory
Linear algebra
Mathematical analysis
Matrices (mathematics)
Matrix algebra
Matrix pencils
Preserver
Singularities
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2022.02.035