Linear and multilinear algebra; matrix theory Definition

Transpose

Builds on Matrix

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.

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Tag	`0W809K6AW0`
URI	`m2://0W809K6AW0`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Transpose
Uses	Matrix

Transpose

Definition

Remarks

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