A note on \(k\)-potence preservers on matrix spaces over complex field
Summary: Let \(\mathbb{C}\) be the field of all complex numbers, \(M_n\) the space of all \(n\times n\) matrices over \(\mathbb{C}\), and \(S_n\) the subspace of \(M_n\) consisting of all symmetric matrices. The map \(\phi:S_n\to M_n\) satisfies that \(A-\lambda B\) is \(k\)-potent in \(S_n\)? implying that \(\phi(A)-\lambda\phi(B)\) is \(k\)-potent in \(M_n\), where \(\lambda\in\mathbb{C}\) then there exist an invertible matrix \(P\in M_n\) and \(\epsilon\in\mathbb{C}\) with \(\epsilon^k=\epsilon\) such that \(\phi(X)=\epsilon P^{-1}(X)P\) for every \(X\in S_n\). Moreover, the inductive method used in this paper can be used to characterise similar maps from \(M_n\) to \(M_n\).
scientific article; zbMATH DE number 6307024
scientific article