Given $N$ matrices $X_k=A_k^T\otimes (B_k\cdot C_0)$ and the vertically concatenated matrix $X = [X_1^T\; X_2^T\; \dots \;X_N^T]^T$, what is the condition for $X$ to have full rank? The matrices $A_k,\;B_k$ and $C_0$ have all full rank and appropriate dimensions. The operator $\otimes$ denotes the krnoecker product.
The rank of $X_k$ can be expressed as $\operatorname{rank}(A_k^T\otimes (B_k\cdot C_0))= \operatorname{rank}(A_k)\cdot \operatorname{rank}(B_k\cdot C_0)$. But what about the rank of the concatenated matrix $X$?