A Study on Compound-Commuting Mappings

Chan, Tai Chong (2021) A Study on Compound-Commuting Mappings. Final Year Project, UTAR.

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Abstract

Let F be a field carrying an involution − and F − be a fixed field of F corresponding to the involution − where F − = {α ∈ F | α = α}. Let m, n be positive integers with m, n > 2. We denote the set of all Hermitian matrices of order n underlying the field F by Hn(F). Furthermore, the (n − 1)-th compound of a matrix A and the rank of the matrix A, we denote them by Cn−1(A) and rk(A), respectively. In our study, we characterise a mapping Υ: Hn(F) → Hm(F) that satisfies one of the following conditions: [P1] Υ(Cn−1(A − B)) = Cm−1(Υ(A) − Υ(B)) for any A, B ∈ Hn(F); [P2] Υ(Cn−1(A+αB)) = Cm−1(Υ(A)+αΥ(B)) for any A, B ∈ Hn(F) and α ∈ F −. In order to obtain a general form of a mapping Υ satisfying [P1] or [P2], we need to impose some assumptions on Υ. If Υ satisfies [P1] with Υ(In) 6= 0m, then Υ satisfies rk(A − B) = n if and only if rk(Υ(A) − Υ(B)) = m for any A, B ∈ Hn(F). Also, if Υ satisfies [P2] with Υ(In) 6= 0m, then Υ is a rank-one non-increasing additive mapping. In case of Υ satisfies [P2] with Υ(In) = 0m, we have Υ(A) = 0m for any A ∈ Hn(F) with rk(A) 6 1, Υ(Cn−1(A)) = 0m for any A ∈ Hn(F) and rk(Υ(A)) 6 m − 2 for any A ∈ Hn(F). Some examples of non-zero mapping Υ satisfying [P2] with Υ(In) = 0m are constructed.

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