There is a $ML \times ML$ matrix expression based on the Kronecker product: $$ \mathbf{J}=\left[ \begin{array}{c} \mathbf{I}_L\otimes \left[ \begin{matrix} 1& 0& \cdots& 0\\ \end{matrix} \right]\\ \mathbf{I}_L\otimes \left[ \begin{matrix} 0& 1& \cdots& 0\\ \end{matrix} \right]\\ \vdots\\ \mathbf{I}_L\otimes \left[ \begin{matrix} 0& 0& \cdots& 1\\ \end{matrix} \right]\\ \end{array} \right] $$ where $\mathbf{I}_L$ is a $L \times L$ identity matrix, $ \left[ 0,0,\cdots,1,0,\cdots,0,0 \right] $ is a $1 \times M$ vector with 1 in the $m$-th column and 0 in the rest.
I would like to know if this matrix $\mathbf{J}$ can be further simplified? So that I can make the calculation more efficient in MATLAB.
Thanks in advance!