# On m-th roots of nilpotent matrices

2021-11-01
A new necessary and sufficient condition for the existence of an m-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the m-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an m-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an m-th root is corrected and generalized. Moreover, for a singular matrix having an m-th root with a pair of nilpotent Jordan blocks of sizes s and l, a new m-th root is constructed by replacing that pair by another one of sizes s + i and 1 - i, for special s, l, i. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix A over an arbitrary field that is a sum of two commuting matrices, several results for the existence of m-th roots of A(k) are obtained.
ELECTRONIC JOURNAL OF LINEAR ALGEBRA

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Citation Formats
S. Öztürk, “On m-th roots of nilpotent matrices,” ELECTRONIC JOURNAL OF LINEAR ALGEBRA, vol. 37, pp. 718–733, 2021, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/95012. 