Idempotence-preserving maps without the linearity and surjectivity assumptions Original Research Article

Let Mn(F) be the space of all n×n matrices over a field F of characteristic not 2, and let Pn(F) be the subset of Mn(F) consisting of all n×n idempotent matrices. We denote by Φn(F) the set of all maps from Mn(F) to itself satisfying A−λBset membership, variantPn(F) if and only if φ(A)−λφ(B)set membership, variantPn(F) for every A,Bset membership, variantMn(F) and λset membership, variantF. It was shown that φset membership, variantΦn(F) if and only if there exists an invertible matrix Pset membership, variantMn(F) such that either φ(A)=PAP−1 for every Aset membership, variantMn(F), or φ(A)=PATP−1 for every Aset membership, variantMn(F). This improved Dolinarʹs result by omitting the surjectivity assumption and extending the complex field to any field of characteristic not 2.