Matrix multiplication

A binary operation on compatible 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 $$ (AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}\ ,\ \ i=1,\ldots,m; j=1,\ldots,p. $$

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 $$ (A \circ B)_{ij} = a_{ij} b_{ij}\ ,\ \ i=1,\ldots,m; j=1,\ldots,n. $$

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