---
tag: MX26N71ZD3
kind: definition
name: Commuting matrices
msc: [15A27]
defines: 8G3X8MPG7E
uses: [0CP7YY6NAG, 076JGP6R3T, 0GYGHKX834]
sources:
  - url: https://en.wikipedia.org/wiki/Commuting_matrices
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] and let $n \ge 1$.
Two $n \times n$ matrices $A$ and $B$ over $R$ *commute* if
$$AB = BA,$$
where the products are [[matrix products|032Q9FB8MT]].
A family of $n \times n$ matrices is *commuting* if each two of its members commute.

## Remarks

Example: $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 5 \\ 0 & 3 \end{pmatrix}$ over $\R$ commute, since $AB = BA = \begin{pmatrix} 3 & 11 \\ 0 & 3 \end{pmatrix}$.
Non-example: $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ do not commute, since $AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ but $BA = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$.
The condition is often written $[A, B] = 0$, where $[A, B] = AB - BA$ is the commutator of $A$ and $B$.
The matrices that commute with a fixed matrix $A$ form the centraliser of $A$ in the matrix ring; it contains every polynomial in $A$.
Over an algebraically closed field, a commuting family of matrices has a common [[eigenvector|0N6960W0BV]].