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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- Dec 11th 2012, 06:26 PMxinglongdadarelatively prime
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,

- Dec 11th 2012, 06:37 PMDrexel28Re: relatively prime
You know that the minimal polynomial is irreducible. So, if was not one, this would say that which would imply that , yeah?

- Dec 11th 2012, 08:51 PMxinglongdadaRe: relatively prime
Why minimal polynomial is irreducible? It of course could be reducible? Do you see the Jordan canonical form of a matrix...

- Dec 11th 2012, 08:57 PMDrexel28Re: relatively prime
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.