On the mth roots of a complex matrix

Abstract

If an n x n complex matrix A is nonsingular, then for every integer m > 1, A has an mth root B, i.e., Bm = A. In this paper, we present a new simple proof for the Jordan canonical form of the root B. 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 Ak (k = 0, 1, 2, . . .).