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

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

where

$$\alpha\equiv p(i-1)+k$$ (2)

$$\beta\equiv q(j-1)+l.$$ (3)

For a $2\times 2$ Matrix A and a $2\times 3$ Matrix B,

$${\hbox{\sf A}}\otimes{\hbox{\sf B}} = \left[\begin{array}{ccc} a_{11}{\hbox{\sf B}}& a_{12}{\hbox{\sf B}}\\ a_{21}{\hbox{\sf B}}& a_{22}{\hbox{\sf B}}\end{array}\right]$$

$$= \left[\begin{array}{ccccccccccccccccccc} a_{11} b_{11} & a_{11} b... ..._{21} b_{31} & a_{21} b_{32} & a_{22} b_{31} & a_{22} b_{32}\end{array}\right].$$