siliciooy0j

2022-03-24

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 $End\left(V\right)\otimes End\left(V\right)$ ?

mhapo933its

Beginner2022-03-25Added 9 answers

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 $n+4\left(\genfrac{}{}{0ex}{}{n}{2}\right)=2{n}^{2}-n$

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 $1,\text{}a,\dots ,{a}^{2n-2}$ 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!

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