Linear and multilinear algebra; matrix theory Definition

Matrix multiplication

Builds on Matrix

Definition

Let $R$ be a ring, let $A = (a_{ij})$ be an $m \times n$ matrix over $R$ , and let $B = (b_{jk})$ be an $n \times p$ matrix over $R$ . The product $AB$ is the $m \times p$ matrix with entries $(AB)_{ik} = \sum_{j=1}^{n} a_{ij} b_{jk}$ for $1 \le i \le m$ and $1 \le k \le p$ .

Remarks

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$ . Column $k$ of $AB$ is the matrix–vector product of $A$ with column $k$ of $B$ . The product is associative and distributive, but it is not commutative. If $A$ and $B$ are the matrices of two linear maps, then $AB$ is the matrix of their composition.

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Matrix definition
Matrix multiplication definition

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Tag	`076JGP6R3T`
URI	`m2://076JGP6R3T`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Matrix multiplication
Uses	Matrix

Matrix multiplication

Definition

Remarks

Reading path

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Further reading

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