ELA, Volume 9, pp. 32-41, April 2002, abstract.  

On the mth roots of a complex matrix

Panayiotis J. Psarrakos

If an nxn complex matrix A is nonsingular, then for
every integer m>1, A has an mth root B, i.e., 
B^m=A. In this paper, a new simple proof for the
Jordan canonical form of the root B is presented. 
Moreover, a necessary and sufficient condition for 
the existence of mth roots of a singular complex 
matrix A is obtained. This condition is in terms
of the dimensions of the null spaces of the 
powers A^k (k =0,1,2...).