Matrices A such that RA=As+1R when Rk=I

This paper examines matrices A∈Cn×n such that RA=As+1R where Rk=I, the identity matrix, and where s and k are nonnegative integers with k⩾2. Spectral theory is used to characterize these matrices. The cases s = 0 and s⩾1 are considered separately since they are analyzed by different techniques.