Let $A\in M_{pq}$ be a given matrix. We want to find matrices $X\in M_p$ and $Y\in M_q$ whose Kronecker product approximates $A$ as well as possible in the Frobenius norm; that is, we want to find solutions to \begin{equation*} \min_{X,Y}||A-(X\otimes Y)||_F^2. \end{equation*} question: Please show that this can be reformulated as a rank-one approximation problem \begin{equation*} \min_{x,y}||B-xy^\ast||_F^2. \end{equation*} where $x$, $y$ are vectors and $B$ is a matrix independent of $x$, $y$. The solutions of the former problem should be given from the solutions of the latter, which can be obtained from the SVD of $B$.
Hints: Notice that there exists a permutation matrix $P$ such that $vec(X\otimes Y ) = P[vec(X) \otimes vec(Y )]$. Also notice that $vec(ab^T ) = b\otimes a$ for vectors $a$, $b$.