Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?



eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $
The three terms commute, whether or not $A$, $B$, and $C$ are symmetric. The eigenvalues are of the form $bc+ac+ab$, where $a$, $b$, and $c$ are eigenvalues of $A$, $B$, and $C$. You just tensor eigenvectors together and you get the answer.


