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I'm currently attempting to write a symmetric matrix with structured blocks into Kronecker-factorized form, but I'm not sure if the task is possible at all. My matrix takes the following form:

$$ M= \begin{bmatrix} a I_1 & \mathbf{0} & gI_1 & \mathbf{0} & iI_1 & \mathbf{0} \\ \mathbf{0} & bI_2 & \mathbf{0} & hI_2 & \mathbf{0} & jI_2 \\ g I_1 & \mathbf{0} & cI_1 & \mathbf{0} & kI_1 & \mathbf{0} \\ \mathbf{0} & hI_2 & \mathbf{0} & dI_2 & \mathbf{0} & lI_2 \\ i I_1 & \mathbf{0} & kI_1 & \mathbf{0} & eI_1 & \mathbf{0} \\ \mathbf{0} & j I_2 & \mathbf{0} & lI_2 & \mathbf{0} & fI_2 \end{bmatrix} $$ where $I_1, I_2$ are identity matrices in $\mathbb{R}^{d_1},\mathbb{R}^{d_2}$ respectively, and $a,b,\cdots,l$ are arbitrary constants. Obviously, it'd be desirable to have written $$ M = W\otimes\begin{bmatrix} I_1 & \mathbf{0} \\ \mathbf{0} & I_2 \end{bmatrix} $$ but I suspect it'd be feasible at all for arbitrary constants. Any feedback/suggestion would be of great help. Thanks!

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To begin, note that there exists a permutation matrix $P$ such that $$ P^TMP = \pmatrix{ aI_1 & gI_1 & iI_1\\ g I_1 & c I_1 & kI_1\\ iI_1 & k I_1 & eI_1\\ &&& bI_2 & h I_2 & j I_2\\ &&& h I_2 & dI_2 & l I_2\\ &&& j I_2 & l I_2 & fI_2 } $$ where the blank spaces denote zeros. We can write each of these blocks as a Kronecker product. That is, we have $$ P^TMP = \pmatrix{\pmatrix{a&g&i\\g&c&k\\i&k&e} \otimes I_1 \\ & \pmatrix{b&h&j\\h&d&l\\j&l&f} \otimes I_2} $$

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