Linear preservers on powers of matrices (Q1183211)

Let \(F\) be a field and \(M_ n(F)\) and \(S_ n(F)\) be the vector spaces of all \(n\times n\) matrices and symmetric matrices over \(F\) respectively. Let \(k\) be a fixed integer, \(k>1\). The characterization of the linear maps \(L\) on \(M_ n(F)\) that satisfy \(L(A^ k)=L(A)^ k\) for all \(A\) when \(\text{char} F=0\text{ or }>k\) is given. A similar result when \(F\) is algebraically closed is obtained for linear maps \(L\) on \(S_ n(F)\). (Actually for \(k=2,3\) the authors need only the weaker hypothesis that \(L\) preserves idempotent and tripotent matrices, respectively to obtain the structure of \(L)\).