Transpose

Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $m \times n$ matrix 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. The matrix of the dual map is the transpose 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.