Linear and multilinear algebra; matrix theory Proposition

Transpose of a product

Builds on Transpose Matrix multiplication

Statement

Let $R$ be a commutative ring. Let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $R$ . Then $(AB)^T = B^T A^T$ , where $^T$ denotes the transpose.

Remarks

The order of the factors reverses. The proof compares entries: both sides have entry $(k, i)$ equal to $\sum_{j} a_{ij} b_{jk}$ . By induction, $(A_1 A_2 \cdots A_k)^T = A_k^T \cdots A_2^T A_1^T$ . If $R$ is not commutative, the identity can fail.

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Matrix definition
Matrix multiplication definition
Transpose definition
Transpose of a product statement

Tag	`09GK5PS824`
URI	`m2://09GK5PS824`
Kind	statement, proposition
Writing	W0 stub
MSC	15A99
Status	This copy of the site has no build data. Run `m2 status 09GK5PS824`.
Uses	Transpose, Matrix multiplication

Transpose of a product

Statement

Remarks

Reading path

Further reading

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