---
tag: 0W809K6AW0
kind: definition
name: Transpose
msc: [15A99]
defines: 09DXWYRD43
uses: [0N1MY1T0W9]
sources:
  - url: https://en.wikipedia.org/wiki/Transpose
    locator: lead section
---

## Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $m \times n$ [[matrix|0VD9ZSWWJG]] over $R$.
The *transpose* of $A$ is the $n \times m$ matrix $A^T$ with entries
$$(A^T)_{ij} = a_{ji}$$
for $1 \le i \le n$ and $1 \le j \le m$.

## Remarks

The transpose of $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ is $\begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$.
The rules $(A^T)^T = A$, $(A + B)^T = A^T + B^T$ and $(cA)^T = c A^T$ hold.
Over a commutative ring, $(AB)^T = B^T A^T$, see [[transpose of a product|09GK5PS824]].
The [[matrix of the dual map is the transpose|0ZPB3180AY]] of the matrix of the map.

## Explanations

### L1 human

To transpose a matrix, flip it over its main diagonal.
Row $1$ becomes column $1$, row $2$ becomes column $2$, and so on.
A $2 \times 3$ matrix becomes a $3 \times 2$ matrix.