Linear transformations on matrices: rank preservers and determinant preservers (Note)
Clemens Lautemann · Linear and Multilinear Algebra · 1981
Characterizations of rank preserving and determinant preserving linear transformations on matrices over an algebraically closed field, due to Marcus and Frobenius respectively, are generalized to linear transformations on matrices over an arbitrary field (rank preservers) or over any field which satisfies a weak cardinality condition (determinant preservers).