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Difference between revisions of "Matrix multiplication"

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(Start article: Matrix multiplication)
 
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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
 
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\,.
+
(AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}, \quad i=1,\ldots,m;\ \ j=1,\ldots,p.
 
$$
 
$$
  
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The Hadamard product, or Schur product, of two $m \times n$ matrices $A$ and  $B$ is the $m \times n$ matrix $AB$ with
 
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\,.
+
(A \circ B)_{ij} = a_{ij} b_{ij}, \quad i=1,\ldots,m;\ \ j=1,\ldots,n.
 
$$
 
$$
  
 
===Kronecker multiplication===
 
===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
+
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},\ \ i=1,\ldots,m\,\ j=1,\ldots,n\,\ k=1,\ldots,p\,\ l=1,\ldots,q\,.
+
(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==
 
==References==
* Gene H. Golub, Charles F. Van Loan, ''Matrix Computations'', Johns Hopkins Studies in the Mathematical Sciences '''3''', JHU Press (2013) ISBN 1421407949
+
* 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
+
* 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
+
* 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}}

Latest revision as of 19:22, 11 November 2023

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

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
How to Cite This Entry:
Matrix multiplication. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_multiplication&oldid=35973