---
tag: 0VTAD25A3G
kind: statement
role: proposition
name: Minimal polynomial divides every annihilating polynomial
msc: [15A18]
uses: [0V58FPM3EQ, 052M4SX57M]
sources:
  - url: https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)
    locator: Formal definition
---

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an endomorphism with [[minimal polynomial|03NDVCSHSB]] $m_T$.
If $p \in F[x]$ and $p(T) = 0$, then $m_T$ divides $p$.

## Remarks

The proof divides with remainder: $p = q m_T + r$ with $\deg r < \deg m_T$.
Then $r(T) = 0$, so $r = 0$ by the minimality of $\deg m_T$.

## Explanations

### L1 human

Every polynomial that kills $T$ is a multiple of the minimal polynomial.
If a matrix satisfies $A^2 = A$, then its minimal polynomial divides $x^2 - x$.
So it is $x$, $x - 1$ or $x^2 - x$.