nilpotent

What it means

1. not comparablering theoryalgebraSuch that, for some positive integer n, xⁿ = 0.

   If a square matrix is upper triangular and has zeros on the diagonal, then it is nilpotent (under the usual matrix multiplication).

   1 more example

   The rest of this book is devoted to determining the conjugacy classes and centralizers of nilpotent elements in L(G) and unipotent elements in G, where G is an exceptional algebraic group of type E₈,E₇, E₆, F₄ or G₂ over an algebraically closed field K of characteristic p. This chapter contains statements of the main results for nilpotent elements.

   2012, Martin W. Liebeck, Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, American Mathematical Society, page 129

2. In any of several technical senses: behaving analogously to nilpotent ring elements as an element of some other algebraic structure; composed of elements displaying such behavior.
   1. not comparableLie theoryalgebraBelonging to the derived algebra of L and such that the adjoint action of x is nilpotent (as a linear transformation on L).
   2. not comparableLie theoryalgebraSuch that the lower central series terminates.

In a sentence

If a square matrix is upper triangular and has zeros on the diagonal, then it is nilpotent (under the usual matrix multiplication).

Where it comes from

From nil (“not any”) + potent (“having power”) with literal meaning “having zero power” - bearing Latin roots nil and potens. Coined in 1870, along with idempotent, by American mathematician Benjamin Peirce to describe elements of associative algebras.