Let $A$ and $B$ be two matrices (not necessarily of the same size). Relative to the lexicographically ordered basis on the space of tensors, the matrix for $Aotimes B$ can be written in block form as follows: if $A=(a_{ij})$, then $$ex Aotimes B=sex{a{ccc} a_{11}B&cdots&a_{1n}B\ vdots&ddots&vdots\ a_{n1}B&cdots&a_{nn}B ea}. eex$$
Solution. Let $Ain scrL(scrH)$, $Bin scrL(scrK)$, and $e_1,cdots,e_n$; $f_1,cdots,f_m$ be the orthonormal basis of $scrH$ and $scrK$ respectively. Then $$eex ea (Aotimes B)(e_iotimes f_j) &=(Ae_i)otimes (Bf_j)\ &=sum_k a_{ki}e_kotimes sum_l b_{lj}f_l\ &=sum_{k,l}a_{ki}b_{lj}e_kotimes f_l\ &=sex{e_1otimes f_1,cdots,e_1otimes f_n,cdots,e_notimes f_n}sex{a{c} a_{1i}b_{1j}\ vdots\ a_{1i}b_{nj}\ vdots\ a_{ni}b_{nj} ea}. eea eeex$$