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Direct Product (Matrix)

Given an $m\times n$ Matrix A and a $p\times q$ Matrix B, their direct product ${\hbox{\sf C}}= {\hbox{\sf A}}\otimes{\hbox{\sf B}}$ is an $(mp)\times(nq)$ Matrix with elements defined by

C_{\alpha\beta} = {\hbox{\sf A}}_{ij} {\hbox{\sf B}}_{kl},
\end{displaymath} (1)

\alpha\equiv p(i-1)+k
\end{displaymath} (2)

\beta\equiv q(j-1)+l.
\end{displaymath} (3)

For a $2\times 2$ Matrix A and a $2\times 3$ Matrix B,
$\displaystyle {\hbox{\sf A}}\otimes{\hbox{\sf B}}$ $\textstyle =$ $\displaystyle \left[\begin{array}{ccc}
a_{11}{\hbox{\sf B}}& a_{12}{\hbox{\sf B}}\\
a_{21}{\hbox{\sf B}}& a_{22}{\hbox{\sf B}}\end{array}\right]$  
  $\textstyle =$ $\displaystyle \left[\begin{array}{ccccccccccccccccccc}
a_{11} b_{11} & a_{11} b...
..._{21} b_{31} & a_{21} b_{32} & a_{22} b_{31} & a_{22} b_{32}\end{array}\right].$  

© 1996-9 Eric W. Weisstein