Let k be a field, V a finite-dimensional



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

Answer & Explanation



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 m1,,mn, and generically MM will have distinct eigenvalues m12,,mn2 with multiplicity one, and m1m2,m1m3,,mn1mn with multiplicity two. Thus the centralizer will have dimension n+4n2=2n2-n
But there are many degenerate cases: for instance if M has eigenvalues 1 1,a,,an1 then MM will have eigenvalues 1, a,,a2n2 with multiplicities 1,2,n1,n,n1,,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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