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Pairs of maps preserving singularity on subsets of matrix algebras

Linear Algebra and its Applications. 2022. Vol. 644. P. 1–27.
A.E. Guterman, A.M. Maksaev, V.V. Promyslov

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) = P AQ for all A ∈ Mn, or of the form T1(A) = T2(A) = P AtQ for all A ∈ Mn, where P, Q ∈ Mn are non-singular matrices.

Research target: Mathematics
Language: English
DOI
Text on another site
Keywords: determinantPreserversMatrix pencilsAutomorphisms of graphs
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