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 $?
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2$\begingroup$ Why was this upvoted? It is an elementary question in linear algebra and not appropriate for the site. Voting to close. $\endgroup$– Qiaochu YuanFeb 11, 2011 at 15:54
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$\begingroup$ I confess, I am the guilty party. Except when I forget, I almost always upvote a question that I answer. If I felt like answering the question, then I figure that the question deserves my vote. $\endgroup$– Greg KuperbergFeb 11, 2011 at 16:02
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$\begingroup$ But note that someone else also upvoted the question. Otherwise, I agree, it's on the easy side. $\endgroup$– Greg KuperbergFeb 11, 2011 at 16:05
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1 Answer
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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.