A localization technique for linear preserver problems

We introduce a new general technique for solving linear preserver problems. The idea is to localize a given linear preserver ϕ at each non-zero vector. In such a way we get vector-valued linear maps on the space of matrices which inherit certain properties from ϕ. If we can prove that such induced maps have a standard form, then ϕ itself has either a standard form or a very special degenerate form. We apply this technique to characterize linear preservers of full rank. Using this technique we further reprove two classical results describing the general form of linear preservers of rank one and linear preservers of the unitary (or orthogonal) group.