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 $l-i$, 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.

Skew-symmetric tridiagonal Bohemian matrices

Image at right: Olga Taussky−Todd in her Caltech office circa 1960, wearing the famous "numbers" dress Abstract: Skew-symmetric tridiagonal Bohemian matrices with population P = [1,i] have eigenvalues with some interesting properties. We explore some of these here, and I prove a theorem showing that the only possible dimensions where nilpotent matrices can occur are one less than a power of two. I explicitly give a set of matrices in this family at dimension m=2ᵏ−1 which are nilpotent, and recursively constructed from those at smaller dimension. I conjecture that these are the only matrices in this family which are nilpotent. This paper will chiefly be of interest to those readers of my prior paper on Bohemian matrices with this structure who want more mathematical details than was provided there, and who want details of what has been proved versus what has been conjectured by experiment. I also give a terrible pun. Don't say you weren't warned.

Nonparallel flat portions on the boundaries of numerical ranges of 4-by-4 nilpotent matrices

The 4-by-4 nilpotent matrices whose numerical ranges have nonparallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained.

Nilpotent Higgs Bundles and the Hodge Metric on the Calabi–Yau Moduli

We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi–Yau moduli.

On the idempotent and nilpotent sum numbers of matrices over certain indecomposable rings and related concepts

We investigate a few special decompositions in arbitrary rings and matrix rings over indecomposable rings into nilpotent and idempotent elements. Moreover, we also define and study the nilpotent sum trace number of nilpotent matrices over an arbitrary ring. Some related notions are explored as well.

Decomposing the Krohn-Rhodes Form of Electroencephalography (EEG) Signals Using Jordan-Chevalley Decomposition Technique

This paper explores how electroencephalography (EEG) signals in the Krohn-Rhodes form can be decomposed further using the Jordan-Chevalley decomposition technique. First, the recorded EEG signals of a seizure were transformed into a set of matrices. Each of these matrices was decomposed into its elementary components using the Krohn-Rhodes decomposition method. The components were then further decomposed into semisimple and nilpotent matrices using the Jordan-Chevalley decomposition. These matrices—which are the extended building blocks of elementary EEG signals—provide evidence that the EEG signals recorded during a seizure contain patterns similar to that of prime numbers.

Representations of ω-Lie algebras and tailed derivations of Lie algebras

We study the representation theory of finite-dimensional [Formula: see text]-Lie algebras over the complex field. We derive an [Formula: see text]-Lie version of the classical Lie’s theorem, i.e., any finite-dimensional irreducible module of a soluble [Formula: see text]-Lie algebra is 1-dimensional (1D). We also prove that indecomposable modules of some 3D [Formula: see text]-Lie algebras could be parametrized by the complex field and nilpotent matrices. We introduce the notion of a tailed derivation of a nonassociative algebra [Formula: see text] and prove that if [Formula: see text] is a Lie algebra, then there exists a one-to-one correspondence between tailed derivations of [Formula: see text] and 1D [Formula: see text]-extensions of [Formula: see text].