Nilpotent

//nɪlˈpəʊtənt// adj, noun

adj, noun ·Uncommon ·Advanced level

Definitions

Noun
  1. 1
    A nilpotent element.

    "The so-called spinor algebra of C(2), the language of the quantum mechanics, is formulated in terms of the idempotents and nilpotents of the geometric algebra of space, including its beautiful representation on the Riemann sphere, and a new proof of the Heisenberg uncertainty principle."

Adjective
  1. 1
    Such that, for some positive integer n, xⁿ = 0. not-comparable

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

  2. 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.; Belonging to the derived algebra of L and such that the adjoint action of x is nilpotent (as a linear transformation on L). not-comparable
  3. 3
    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.; Such that the lower central series terminates. not-comparable
  4. 4
    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.; Admitting a central series of finite length. not-comparable
  5. 5
    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.; Such that there exists a natural number k with Iᵏ = 0. not-comparable
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  1. 6
    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.; Containing only nilpotent elements. not-comparable
  2. 7
    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.; Such that there exists some natural number n (called the index of the algebra) such that all products (of elements in the given algebra) of length n are zero. not-comparable
Adjective
  1. 1
    equal to zero when raised to a certain power wordnet

Synonyms

All synonyms
idempotent

Example

More examples

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

Etymology

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.

Related phrases

nilpotent algebranilpotent idealnilpotent orbitnilpotent semigroup

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Data sourced from Wiktionary, WordNet, CMU, and other open linguistic databases. Updated March 2026.