Permanent Polya problem for additive surjective maps

A. E. Guterman; I. A. Spiridonov

doi:10.1016/j.laa.2020.04.001

Permanent Polya problem for additive surjective maps

A. E. Guterman, I. A. Spiridonov

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let M_n(F) denote the set of square matrices of size n over a field F with characteristics different from two. We say that the map f:M_n(F)→M_n(F) is additive if f(A+B)=f(A)+f(B) for all A,B∈M_n(F). The main goal of this paper is to prove that for n>2 there are no additive surjective maps T:M_n(F)→M_n(F) such that per(T(A))=det⁡(A) for all A∈M_n(F). Also we show that an arbitrary additive surjective map T:M_n(F)→M_n(F) which preserves permanent is linear and thus can be completely characterized.

Original language	English
Pages (from-to)	140-155
Number of pages	16
Journal	Linear Algebra and Its Applications
Volume	599
DOIs	https://doi.org/10.1016/j.laa.2020.04.001
State	Published - 15 Aug 2020
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2020 Elsevier Inc.

Funding

This work was financially supported by the Russian Science Foundation under grant 16-11-10075 .

Funders	Funder number
Russian Science Foundation	16-11-10075

Keywords

Additive maps
Determinant
Linear maps
Permanent

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10.1016/j.laa.2020.04.001

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title = "Permanent Polya problem for additive surjective maps",

abstract = "Let Mn(F) denote the set of square matrices of size n over a field F with characteristics different from two. We say that the map f:Mn(F)→Mn(F) is additive if f(A+B)=f(A)+f(B) for all A,B∈Mn(F). The main goal of this paper is to prove that for n>2 there are no additive surjective maps T:Mn(F)→Mn(F) such that per(T(A))=det⁡(A) for all A∈Mn(F). Also we show that an arbitrary additive surjective map T:Mn(F)→Mn(F) which preserves permanent is linear and thus can be completely characterized.",

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AU - Spiridonov, I. A.

PY - 2020/8/15

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N2 - Let Mn(F) denote the set of square matrices of size n over a field F with characteristics different from two. We say that the map f:Mn(F)→Mn(F) is additive if f(A+B)=f(A)+f(B) for all A,B∈Mn(F). The main goal of this paper is to prove that for n>2 there are no additive surjective maps T:Mn(F)→Mn(F) such that per(T(A))=det⁡(A) for all A∈Mn(F). Also we show that an arbitrary additive surjective map T:Mn(F)→Mn(F) which preserves permanent is linear and thus can be completely characterized.

AB - Let Mn(F) denote the set of square matrices of size n over a field F with characteristics different from two. We say that the map f:Mn(F)→Mn(F) is additive if f(A+B)=f(A)+f(B) for all A,B∈Mn(F). The main goal of this paper is to prove that for n>2 there are no additive surjective maps T:Mn(F)→Mn(F) such that per(T(A))=det⁡(A) for all A∈Mn(F). Also we show that an arbitrary additive surjective map T:Mn(F)→Mn(F) which preserves permanent is linear and thus can be completely characterized.

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KW - Determinant

KW - Linear maps

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ER -

Permanent Polya problem for additive surjective maps

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