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Pairs of maps preserving singularity on subsets of matrix algebras
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NSTL
Elsevier
Let F be an algebraically closed field and M-n be the n x 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 + lambda B. In particular, we characterize maps T-1, T-2: M-n & nbsp;->& nbsp;M-n satisfying the condition A + lambda B is singular if and only if T-1(A) + lambda T-2(B) is singular, for any A, B is an element of & nbsp;M-n and any non-zero lambda is an element of & nbsp;F. Namely, we prove that in this case T-1 = T-2 and they are of the form T-1(A) = T-2(A) = PAQ for all A is an element of & nbsp;M-n, or of the form T-1(A) = T-2(A) = PA(t)Q for all A is an element of & nbsp;M-n, where P, Q is an element of & nbsp;M-n are non-singular matrices. (C)& nbsp;2022 Elsevier Inc. All rights reserved.(c) 2022 Elsevier Inc. All rights reserved.
DeterminantPreserverAutomorphisms of graphsMatrix pencilsTOTAL GRAPHINVERTIBILITY
NSTL
Elsevier
