Linear maps on Mn(C) preserving the local spectrum

Let x0∈Cn be a nonzero vector. We prove that if a linear map φ:Mn(C)→Mn(C) preserves the local spectrum at x0; i.e., σT(x0)=σφ(T)(x0) for all T∈Mn(C), then there exists an invertible matrix A such that A(x0)=x0 and φ(T)=ATA-1 for every T.