# Sum of Kronecker Products, $(\mathbf I_n\otimes\mathbf I_m)+(\mathbf A\otimes \mathbf B)$

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.

• Isn't $I\otimes I=I$? – Omnomnomnom Oct 11 '16 at 23:33
• Yes, in this case $(\mathbf I_n \otimes \mathbf I_m)=\mathbf I_{nm}$. – mzp Oct 11 '16 at 23:35

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 $B=\mu I_m$.
• It admits an infinity of solutions; there is clearly a degree of freedom in the relative scale of $C$ and $D$. – Sheljohn Oct 12 '16 at 1:14