On triangularizability of the commutant of a single matrix

The main purpose of this paper is to characterize triangularizable matrices A ∈ Mn(F) whose commutants are triangularizable, where F is an arbitrary field. More precisely, we show that the commutant of a triangularizable matrix A ∈ Mn(F) is triangularizable if and only if for any eigenvalue λ of A, the corresponding Jordan blocks in the Jordan canonical form of A have distinct sizes.