Linear Bijective Maps Sending Nilpotent Matrices into Matrices of Local Spectral Radius Zero at a Fixed Vector

For a natural number n , denote by Mₙ M n the algebra of all matrices of size n over the complex field C C . Let also x₀ x 0 be a fixed nonzero vector in Cⁿ C n . In this paper, we characterize linear bijective maps on Mₙ M n sending nilpotent matrices into matrices of local spectral radius zero at x₀ x 0 . As a corollary, we also give the general form of linear bijective maps on Mₙ M n sending matrices of local spectral radius zero at x₀ x 0 into matrices of the same type.