# Solution space of Kronecker product of variables

i want to solve the following equation. $\begin{bmatrix} \mathbf{A_1} & \mathbf{A_2} & \mathbf{A_3} & \mathbf{A_4} \\ \mathbf{A_5} & \mathbf{A_6} & \mathbf{A_7} & \mathbf{A_8} \end{bmatrix} \begin{bmatrix} \mathbf{w_1} \\ \mathbf{w_2} \\ \mathbf{w_3} \\ \mathbf{w_4}\end{bmatrix} = \mathbf{0}$ say Equation (1)

where $\mathbf{A}_i = \mathbf{B}_i \otimes \mathbf{D}_i$. Here, $\mathbf{B}_i$ and $\mathbf{D}_i$ are matrice of size 2 x 2 with non-zero elements and full rank. These matrices are given.

$\mathbf{w}_i = \mathbf{x}_i \otimes \mathbf{y}_i$. Here, $\mathbf{x}_i$ and $\mathbf{y}_i$ are vectors of size 2 x 1 and are variables. How to find the vectors $\mathbf{x}_i$ and $\mathbf{y}_i$?

Note: I calculated the solution space of the $\mathbf{w}$ vectors by calculating the null space of the matrix. i took one vector from the solution space and tried to find $\mathbf{x}_i$ and $\mathbf{y}_i$ from $\mathbf{w}_i$ but it didnot satisfy $\mathbf{w}_i = \mathbf{x}_i \otimes \mathbf{y}_i$.

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Your matrix is $8\times 16$ and therefore the nullspace is at least 8 dimensional but there is no guarantee that the overall rank is not less than 8. Did you check the overall rank? How did you calculate the $x_i$'s and $y_i$'s? –  user13838 Sep 16 '11 at 15:40
i checked the rank of 8 x 16 matrix, it is 8 and hence the null space is of rank 8. –  karthik Sep 16 '11 at 15:46
i checked the rank of 8 x 16 matrix, it is 8 and hence the null space is of rank 8. i took $\mathbf{w}_1$ and and reversed the kronecker operation. That is i took the four elements of $\mathbf{w}_1$ and divided the first two elements with the third and fourth elements, element wise to check if both division results in a constant. I thinks kronecker produce is not uniquely reversible but to a scaling factor. –  karthik Sep 16 '11 at 15:52
–  karthik Sep 16 '11 at 16:05

I have previously overlooked at the the following problem and thought that it is solvable but apparently I can't without using some nonlinear optimization code. Based on our discussion, I might have mislead @karthik. Hence, I have deleted my previous comments. And posting this answer so that, someone else doesn't get confused.

Let $x_i,y_i \in\mathbb{R}^2,w_i\in\mathbb{R}^4$ for $i=1,\ldots,4$. Given a $16\times 8$ real valued full rank matrix N, how can we find an arbitrary solution $r\in\mathbb{R}^8, x_i$ and $y_i$ such that $$Nr = \begin{pmatrix}x_1\otimes y_1\\ x_2\otimes y_2 \\ x_3\otimes y_3 \\x_4\otimes y_4\end{pmatrix}$$ holds?

Mistakenly, I thought this is can be brought down to a matrix equation with a invertible matrix variable.

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could you give some hint on how you solved the above problem using nonlinear optimization methods - what kind of algebra or mathematics shall i look reg. this. Also, the entries are complex values - i think this do not change the approach much. –  karthik Sep 16 '11 at 20:44
Well I solved it via branch and bound techniques with some custom code. I don't know if I can give it to you because it belongs to somebody else. I will try to find an open source one or MATLAB's solvers. In the mean time try to look at Yalmip. It is a wonder when it comes to parsing optimization problems. –  user13838 Sep 16 '11 at 20:56
Thanks for the hints! –  karthik Sep 21 '11 at 11:46
You're welcome, I couldn't do much but good luck. –  user13838 Sep 21 '11 at 12:39