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Suppose $\mathbf A$ is $n\times n$ and $\mathbf B$ is $m\times m$. Is it possible to write $$ (\mathbf I_n\otimes\mathbf I_m)+(\mathbf A\otimes \mathbf B)$$ as a Kronecker product between two matrices?

Obs.: The answer to this question is not exactly applicable, but seems to be related.

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  • $\begingroup$ Isn't $I\otimes I=I$? $\endgroup$ – Omnomnomnom Oct 11 '16 at 23:33
  • $\begingroup$ Yes, in this case $(\mathbf I_n \otimes \mathbf I_m)=\mathbf I_{nm}$. $\endgroup$ – mzp Oct 11 '16 at 23:35
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As explained in the answer to the question you cited, the equation: $$ C\otimes D = I_n\otimes I_m + A\otimes B $$ where $C$ and $D$ are unknowns, admits solutions iff $A=\lambda I_n$ and $B=\mu I_m$.

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  • $\begingroup$ I understood it as saying that it admits a unique solution iff these conditions hold. $\endgroup$ – mzp Oct 12 '16 at 0:47
  • $\begingroup$ But I think I was wrong before, the other answer to the question does appear to be applicable. Thank you for your help. $\endgroup$ – mzp Oct 12 '16 at 0:49
  • $\begingroup$ It admits an infinity of solutions; there is clearly a degree of freedom in the relative scale of $C$ and $D$. $\endgroup$ – Sheljohn Oct 12 '16 at 1:14

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