Given two matrices $\displaystyle A=\{A_{ij}\}_{n \times m} , B=\{B_{kl}\}_{P \times Q} $, how do i define the direct product of $\displaystyle A \otimes B $?

I understand what it means, $\displaystyle (A \otimes B)_{ij} = A_{ij}B$ but how do i define it using the correct notation?

Thanks