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

We study matrices A∈Cn×n such that As+1R=RA⁎ where Rk=In, and s,k are nonnegative integers with k≥2; such matrices are called {R,s+1,k,⁎}-potent matrices. The s=0 case corresponds to matrices such that A=RA⁎R−1 with Rk=In, and is studied using spectral properties of the matrix R. For s≥1, various characterizations of the class of {R,s+1,k,⁎}-potent matrices and relationships between these matrices and other classes of matrices are presented.