Pairs of maps preserving singularity on subsets of matrix algebras

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

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Pages (from-to)1-27
Number of pages27
JournalLinear Algebra and Its Applications
StatePublished - 1 Jul 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.


  • Automorphisms of graphs
  • Determinant
  • Matrix pencils
  • Preserver


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