 # Let k be a field, V a finite-dimensional k-vectorspace and tobeint39r 2022-04-30 Answered
Let k be a field, V a finite-dimensional k-vectorspace and $M\in End\left(V\right)$. How can I determine Z, the centralizer of $M\otimes M$ in $End\left(V\right)\otimes End\left(V\right)$?
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Step 1
This looks like it can get complicated. Generically, at least over an algebraically closed field, a matrix M will have distinct eigenvalues ${m}_{1},\dots ,{m}_{n}$, and generically $M\otimes M$ will have distinct eigenvalues ${m}_{1}^{2},\dots ,{m}_{n}^{2}$ with multiplicity one, and ${m}_{1}{m}_{2},{m}_{1}{m}_{3},\dots ,{m}_{n-1}{m}_{n}$ with multiplicity two. Thus the centralizer will have dimension $n+4\left(\genfrac{}{}{0}{}{n}{2}\right)=2{n}^{2}-n$
But there are many degenerate cases: for instance if M has eigenvalues 1 $1,a,\dots ,{a}^{n-1}$ then $M\otimes M$ will have eigenvalues  with multiplicities $1,2,\dots n-1,n,n-1,\dots ,1$. Things can get more complicated still.
Then M might have non-trivial Jordan blocks, and then the real fun starts!