ON M-TH ROOTS OF NILPOTENT MATRICES


ÖZTÜRK S.

ELECTRONIC JOURNAL OF LINEAR ALGEBRA, vol.37, pp.718-733, 2021 (Journal Indexed in SCI) identifier

  • Publication Type: Article / Article
  • Volume: 37
  • Publication Date: 2021
  • Title of Journal : ELECTRONIC JOURNAL OF LINEAR ALGEBRA
  • Page Numbers: pp.718-733
  • Keywords: Jordan canonical form, Roots of nilpotent matrices, Rootless matrices, System of linear Diophantine equations, Nonnegative integer solutions

Abstract

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.