On m-th roots of nilpotent matrices

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.


On the arithmetic complexity of Strassen-like matrix multiplications
Cenk, Murat (2017-05-01)
The Strassen algorithm for multiplying 2 x 2 matrices requires seven multiplications and 18 additions. The recursive use of this algorithm for matrices of dimension n yields a total arithmetic complexity of (7n(2.81) - 6n(2)) for n = 2(k). Winograd showed that using seven multiplications for this kind of matrix multiplication is optimal. Therefore, any algorithm for multiplying 2 x 2 matrices with seven multiplications is called a Strassen-like algorithm. Winograd also discovered an additively optimal Stras...
On the computation of generalized division polynomials
Küçüksakallı, Ömer (2015-01-01)
We give an algorithm to compute the generalized division polynomials for elliptic curves with complex multiplication. These polynomials can be used to generate the ray class fields of imaginary quadratic fields over the Hilbert class field with no restriction on the conductor.
Almost periodic solutions of the linear differential equation with piecewise constant argument
Akhmet, Marat (2009-10-01)
The paper is concerned with the existence and stability of almost periodic solutions of linear systems with piecewise constant argument where t∈R, x ∈ Rn [·] is the greatest integer function. The Wexler inequality [1]-[4] for the Cauchy's matrix is used. The results can be easily extended for the quasilinear case. A new technique of investigation of equations with piecewise argument, based on an integral representation formula, is proposed. Copyright © 2009 Watam Press.
Öztürk, Semra (2018-01-01)
Let A and B be matrices which are polynomials in r pairwise commuting nilpotent matrices over a field. We give a sufficient condition for the null space of A(i) to equal that of B-i for all i, in particular, for A and B to be similar.
A note on the minimal polynomial of the product of linear recurring sequences
Cakcak, E (1999-08-06)
Let F be a field of nonzero characteristic, with its algebraic closure, F. For positive integers a, b, let J(a, b) be the set of integers k, such that (x - 1)k is the minimal polynomial of the termwise product of linear recurring sequences sigma and tau in F ($) over bar, with minimal polynomials (x - 1)(a) and (x - 1)(b) respectively. This set plays a crucial role in the determination of the product of linear recurring sequences with arbitrary minimal polynomials. Here, we give an explicit formula to deter...
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.