---
tag: 052M4SX57M
kind: definition
name: Annihilating polynomial
msc: [15A18]
defines: 0PBN8VEKY8
uses: [0BNEXSKNRV]
sources:
  - url: https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)
    locator: Formal definition
---

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism.
A polynomial $p \in F[x]$ *annihilates* $T$ when $p(T) = 0$, where $p(T)$ is the [[value of $p$ at $T$|0E3MV6E6MG]].
Such a polynomial is an *annihilating polynomial* of $T$.

## Remarks

The annihilating polynomials of $T$ form an ideal of $F[x]$.
If $\dim V = n$ is finite, then a nonzero annihilating polynomial exists, because the $n^2 + 1$ maps $\id_V, T, \dots, T^{n^2}$ are linearly dependent.
The monic generator of this ideal is the [[minimal polynomial|03NDVCSHSB]] of $T$.
For a [[projection|033DA1WE8Y]] $P$, the polynomial $x^2 - x$ annihilates $P$.

## Explanations

### L1 human

An annihilating polynomial is a polynomial that becomes the zero map when we put $T$ into it.
For example, a reflection $R$ satisfies $R^2 = I$, so $x^2 - 1$ annihilates it.
Such relations tell us a lot about the possible eigenvalues.