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

===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 $\alpha\beta : R^m \rightarrow R^p$.

===Hadamard multiplication===
The Hadamard product, or Schur product, of two $m \times n$ matrices $A$ and  $B$ is the $m \times n$ matrix $AB$ with
$$
(A \circ B)_{ij} = a_{ij} b_{ij}, \quad i=1,\ldots,m;\ \  j=1,\ldots,n.
$$

===Kronecker multiplication===
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
$$
(A \otimes B)_{(i-1)p+k,(j-1)q+l} = a_{ij} b_{kl}, \quad i=1,\ldots,m;\ \ j=1,\ldots,n;\ \ k=1,\ldots,p;\ \ l=1,\ldots,q.
$$

==References==
* Gene H. Golub, Charles F. Van Loan, ''Matrix Computations'', Johns Hopkins Studies in the Mathematical Sciences '''3''', JHU Press (2013) ISBN 1421407949
* James E. Gentle, ''Matrix Algebra: Theory, Computations, and Applications in Statistics'', Springer Texts in Statistics, Springer (2007) ISBN 0-387-70872-3
* Manfred Schroeder, ''Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity'', Springer (2008) ISBN 3-540-85297-2