---
tag: 093X6MN6Y6
kind: statement
role: corollary
name: Minimal polynomial divides the characteristic polynomial
msc: [15A18]
uses: [0VTAD25A3G, 0F765KFC2J]
sources:
  - url: https://en.wikipedia.org/wiki/Minimal_polynomial_(linear_algebra)
    locator: Properties
---

## Statement

Let $F$ be a field and let $A$ be an $n \times n$ matrix over $F$.
Then the [[minimal polynomial|03NDVCSHSB]] of $A$ divides the [[characteristic polynomial|0VSSPAFM2C]] of $A$.
In particular, $\deg m_A \le n$.

## Remarks

The [[Cayley–Hamilton theorem|0F765KFC2J]] gives $p_A(A) = 0$, and the minimal polynomial divides each annihilating polynomial.
The characteristic polynomial divides a power of the minimal polynomial, because both have the same irreducible factors.
For $I_2$, $m = t - 1$ divides $p = (t - 1)^2$ and is not equal to it.

## Explanations

### L1 human

The minimal polynomial is a divisor of the characteristic polynomial.
So its degree is at most the size of the matrix.
Both polynomials have the same roots.