Open AccessThe algebra generated by nilpotent elements in a matrix centralizer
Author(s) -
Ralph John de la Cruz,
Eloise Misa
Publication year - 2021
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2022.6503
Subject(s) - centralizer and normalizer , subalgebra , mathematics , nilpotent , nilpotent matrix , pure mathematics , cartan subalgebra , square matrix , matrix (chemical analysis) , set (abstract data type) , algebra over a field , discrete mathematics , combinatorics , eigenvalues and eigenvectors , symmetric matrix , affine lie algebra , computer science , physics , chemistry , current algebra , chromatography , quantum mechanics , programming language
For an arbitrary square matrix $S$, denote by $C(S)$ the centralizer of $S$, and by $C(S)_N$ the set of all nilpotent elements in $C(S)$. In this paper, we use the Weyr canonical form to study the subalgebra of $C(S)$ generated by $C(S)_N$. We determine conditions on $S$ such that $C(S)_N$ is a subalgebra of $C(S)$. We also determine conditions on $S$ such that the subalgebra generated by $C(S)_N$ is $C(S).$