## 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 T_{1},T_{2}:M_{n}→M_{n} satisfying the condition A+λB is singular if and only if T_{1}(A)+λT_{2}(B) is singular, for any A,B∈M_{n} and any non-zero λ∈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∈M_{n}, or of the form T_{1}(A)=T_{2}(A)=PA^{t}Q for all A∈M_{n}, where P,Q∈M_{n} are non-singular matrices.

Original language | English |
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Pages (from-to) | 1-27 |

Number of pages | 27 |

Journal | Linear Algebra and Its Applications |

Volume | 644 |

DOIs | |

State | Published - 1 Jul 2022 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2022 Elsevier Inc.

## Keywords

- Automorphisms of graphs
- Determinant
- Matrix pencils
- Preserver