## Abstract

Let Mat_{n}(F) denote the set of square n×n matrices over a field F of characteristic different from two. The permanental rank prk(A) of a matrix A∈Mat_{n}(F) is the size of the maximal square submatrix in A with nonzero permanent. By Λ^{k} and Λ^{≤k} we denote the subsets of matrices A∈Mat_{n}(F) with prk(A)=k and prk(A)≤k, respectively. In this paper for each 1≤k≤n−1 we obtain a complete characterization of linear maps T:Mat_{n}(F)→Mat_{n}(F) satisfying T(Λ^{≤k})=Λ^{≤k} or bijective linear maps satisfying T(Λ^{≤k})⊆Λ^{≤k}. Moreover, we show that if F is an infinite field, then Λ^{k} is Zariski dense in Λ^{≤k} and apply this to describe such bijective linear maps satisfying T(Λ^{k})⊆Λ^{k}.

Original language | English |
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Pages (from-to) | 325-340 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 680 |

DOIs | |

State | Published - 1 Jan 2024 |

### Bibliographical note

Publisher Copyright:© 2023 Elsevier Inc.

### Funding

The second author is supported by ISF Moked grant 2919/19.

Funders | Funder number |
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Israel Science Foundation | 2919/19 |

## Keywords

- Linear map
- Permanent
- Preservers
- Rank