# Kronecker Product Reformulation

I have 2 questions concerning the reformulation of a Kronecker products where i kind of got stuck. Firstly assume the matrix $V\in \mathbb{C}^{N\times M}$ is defined as $$V=\left[\begin{matrix} v_0 \\ \vdots\\ v_{N-1}\\ \end{matrix}\right]$$ with $v_i \in\mathbb{C}^{1\times M}, i\in\{0, \ldots,N-1\}$. Furthermore let vector $\Gamma\in\mathbb{C}^{MP \times 1}$ and we want to calculate $$B=\left[\begin{matrix} I_P\otimes v_0 \\ \vdots\\ I_P \otimes v_{N-1}\\ \end{matrix}\right]\Gamma$$ Is it possible to reformulate B in terms of $V$?.

Using this I actually want to calculate the following product $$C = D^H(I_N\otimes f)\left[\begin{matrix} I_P\otimes v_0 \\ \vdots\\ I_P \otimes v_{N-1}\\ \end{matrix}\right]\Gamma$$ with $D\in\mathbb{C}^{N\times L}$ and $L\leq N$ and $f\in\mathbb{C}^{1\times P}$. Can we reformulate $C$ in such a way that $D^H$ is multiplied with $V$ to obtain a matrix $V'=D^HV$ to obtain something like $$C = (I_?\otimes f)\left[\begin{matrix} I_?\otimes v'_0 \\ \vdots\\ I_? \otimes v'_{N-1}\\ \end{matrix}\right]\Gamma$$ so the matrix D is directly applied? Could you also maybe provide some resources references where I can look into further detail? It would really help me!

Thank you a lot!