Let k be a field, V a finite-dimensional k-vectorspace and M∈End(V). How can I determine Z, the centralizer of M⊗M in End(V)⊗End(V)?

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mhapo933its · Accepted Answer

Step 1This looks like it can get complicated. Generically, at least over an algebraically closed field, a matrix M will have distinct eigenvalues&amp;nbsp;m1,…,mn, and generically&amp;nbsp;M⊗M&amp;nbsp;will have distinct eigenvalues&amp;nbsp;m12,…,mn2&amp;nbsp;with multiplicity one, and&amp;nbsp;m1m2,m1m3,…,mn−1mn&amp;nbsp;with multiplicity two. Thus the centralizer will have dimension&amp;nbsp;n+4n2=2n2-nBut there are many degenerate cases: for instance if M has eigenvalues 1&amp;nbsp;1,a,…,an−1&amp;nbsp;then&amp;nbsp;M⊗M&amp;nbsp;will have eigenvalues&amp;nbsp;1,&amp;nbsp;a,…,a2n−2&amp;nbsp;with multiplicities&amp;nbsp;1,2,…n−1,n,n−1,…,1. Things can get more complicated still.Then M might have non-trivial Jordan blocks, and then the real fun starts!

Let k be a field, V a finite-dimensional

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