How to rewrite kronecker products as linear combination of matrices

Question

Is it possible to write a matrix $U$ such that $U \otimes U \approx A\Lambda A^T$, and $U \approx AB\Lambda_1B^TA^T$? Here $U \in R^{n \times n}$, $A \in R^{n \times k}$, $B \in R^{k \times k_1}$, and $\Lambda_1$ and $\Lambda$ are diagonal matrices. Also $n > k > k_1$. I am not sure if I can remove approx in above equations with equality.

Answer 1

No, not really. If $U$ is $n \times n$ then $U \otimes U$ is $n^2 \times n^2$. Therefore, $A$ has $n^2$ rows according to your first equation, which means it cannot be used to approximate $U$ in your first equation.