Assume that there is a $NL \times K$ matrix $G=\begin{bmatrix} G_1 \\ G_2 \\ \vdots \\ G_N \end{bmatrix}$ such that $rank(G)=K$ and another $NM \times Q$ matrix $H=\begin{bmatrix} H_1 \\ H_2 \\ \vdots \\ H_N \end{bmatrix}$ such that $rank(H)=Q$.
Now consider the matrix $Z=\begin{bmatrix} G_1 \otimes H_1 \\ G_2 \otimes H_2 \\ \vdots \\ G_N \otimes H_N \end{bmatrix}$. Can we say that $rank(Z)=KQ$? If so how to prove this?