Let be a linear transform on a vector field . Suppose is the minimal polynomial of , is a polynomial. Show that if is invertible, then is relatively prime. That is,
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You know that the minimal polynomial is irreducible. So, if was not one, this would say that which would imply that , yeah?
Why minimal polynomial is irreducible? It of course could be reducible? Do you see the Jordan canonical form of a matrix...
My mistake, I was being silly. The key is that the eigenvalues of are where are the eigenvalues of . Now, note that since is invertible it has no zero eigenvalues, and so for all --I think you can conclude from that.
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