Matrices A such that R A = A^{s + 1} R when R^k = I

This paper examines matrices A∈C^(n×n) such that R A = A^(s+1) R where R^k = 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.