Determinants preserving maps on the spaces of symmetric matrices and skew-symmetric matrices

2021/03/22 by Ratsiri Sanguanwong, Sanguanwong, Ratsiri, Kijti Rodtes +1
Engineering · Mathematics · #15A15 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2103.11725

openalex publication_date 2021/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Denote Σ_n and Q_n the set of all n × n symmetric and skew-symmetric matrices over a field \mathbbF, respectively, where char(\mathbbF)≠ 2 and | \mathbbF | ≥ n²+1. A characterization of ϕ,ψ:Σ_n → Σ_n, for which at least one of them is surjective, satisfying det(ϕ(x)+ψ(y))=det(x+y) (x,y∈ Σ_n) is given. Furthermore, if n is even and ϕ,ψ:Q_n → Q_n, for which ψ is surjective and ψ(0)=0, satisfy det(ϕ(x)+ψ(y))=det(x+y) (x,y∈ Q_n), then ϕ=ψ and ψ must be a bijective linear map preserving the determinant.

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