Kronecker Product and state space

Question

I am reading a paper, and one step of it seems like the following:

If $S_1 = $

Then, $I \otimes S_1$ =

How to show it? (Suppose dimension of all of them are correct. $I$ is the identity matrix.)

Is it possible to replace $I$ to other matrix with the same dimension?

Answer 1

According to rhe Wikipedia definition $$ I \otimes S = [\delta_{ij} S] = \left[ \begin{matrix} S & 0 & \cdots & 0 \\ 0 & S & \cdots& 0 \\ \vdots & \vdots& \ddots & \vdots\\ 0 & 0 &\cdots & S \end{matrix} \right] $$ which is not what you want to show.

Maybe the product is defined the other way round? $$ I \, \hat{\otimes}\, S = S \otimes I = [s_{ij} I] = \left[ \begin{array}{c|c} a_{ij} I & b_{ij} I \\ \hline c_{ij} I & d_{ij} I \\ \end{array} \right] = \left[ \begin{array}{c|c} A \otimes I & B \otimes I \\ \hline C \otimes I & D \otimes I \\ \end{array} \right] = \left[ \begin{array}{c|c} I \,\hat{\otimes}\, A & I \,\hat{\otimes}\, B \\ \hline I \,\hat{\otimes}\, C & I \,\hat{\otimes}\, D \\ \end{array} \right] $$