# Matrix multiplication

A binary operation on compatible matrices over a ring $R$. There are several such operations.

## Contents

### Cayley multiplication

Most usually what is referred to as "matrix multiplication". The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $AB$ with entries $$(AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}, \quad i=1,\ldots,m;\ \ j=1,\ldots,p.$$

The multiplication corresponds to composition of linear maps. If $A$ is the matrix of a linear map $\alpha : R^m \rightarrow R^n$ and $B$ is the matrix of a linear map $\beta : R^n \rightarrow R^p$, then $AB$ is the matrix of the linear map $\alpha\beta : R^m \rightarrow R^p$.

The Hadamard product, or Schur product, of two $m \times n$ matrices $A$ and $B$ is the $m \times n$ matrix $AB$ with $$(A \circ B)_{ij} = a_{ij} b_{ij}, \quad i=1,\ldots,m;\ \ j=1,\ldots,n.$$
The Kronecker product, also tensor product or direct product, of an $m \times n$ matrix $A$ and an $p \times q$ matrix $B$ is the $mp \times nq$ matrix $AB$ with entries $$(A \otimes B)_{(i-1)p+k,(j-1)q+l} = a_{ij} b_{kl}, \quad i=1,\ldots,m;\ \ j=1,\ldots,n;\ \ k=1,\ldots,p;\ \ l=1,\ldots,q.$$