[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.4.6

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 $A\otimes B$ can be written in block form as follows: if $A=(a_{ij})$, then $$\bex A\otimes B=\sex{\ba{ccc} a_{11}B&\cdots&a_{1n}B\\ \vdots&\ddots&\vdots\\ a_{n1}B&\cdots&a_{nn}B \ea}. \eex$$

Solution. Let $A\in \scrL(\scrH)$, $B\in \scrL(\scrK)$, and $e_1,\cdots,e_n$; $f_1,\cdots,f_m$ be the orthonormal basis of $\scrH$ and $\scrK$ respectively. Then $$\beex \bea (A\otimes B)(e_i\otimes f_j) &=(Ae_i)\otimes (Bf_j)\\ &=\sum_k a_{ki}e_k\otimes \sum_l b_{lj}f_l\\ &=\sum_{k,l}a_{ki}b_{lj}e_k\otimes f_l\\ &=\sex{e_1\otimes f_1,\cdots,e_1\otimes f_n,\cdots,e_n\otimes f_n}\sex{\ba{c} a_{1i}b_{1j}\\ \vdots\\ a_{1i}b_{nj}\\ \vdots\\ a_{ni}b_{nj} \ea}. \eea \eeex$$