A matrix equation $X^n = aI$
  In this paper, we study a matrix equation $X^n = aI$. We factorize $X^n - aI$ based upon the factorization of $x^n - a$ and then give a necessary and sufficient condition for one of the factors to be the zero matrix. 
Taehyeok Heo, Jihoon Choi, and Suh-Ryung Kim
math.RA