Commutators, matrices and an identity of Copeland

Darij Grinberg

Commutators, matrices and an identity of Copeland

Rings and Algebras 2019-08-27 v1 Combinatorics

Authors: Darij Grinberg

Abstract

Given two elements $a$ and $b$ of a noncommutative ring, we express $\left( ba\right)^n$ as a "row vector times matrix times column vector" product, where the matrix is the $n$ -th power of a matrix with entries $\dbinom{i}{j}\operatorname{ad}_a^{i-j}\left( b\right)$ . This generalizes a formula by Tom Copeland used in the study of Pascal-style matrices.

Keywords

commutators noncommutative ring matrix powers copeland identity pascal-style matrices

Cite

@article{arxiv.1908.09179,
  title  = {Commutators, matrices and an identity of Copeland},
  author = {Darij Grinberg},
  journal= {arXiv preprint arXiv:1908.09179},
  year   = {2019}
}

Comments

30 pages. No plans to submit to journals. Partly expository (the proofs are sufficiently elementary and detailed to be readable at Abstract Algebra I level). Posted here for the sake of referencing from the OEIS

Commutators, matrices and an identity of Copeland

Abstract

Keywords

Cite

Comments

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