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The Cayley Hamilton theorem states for a transformation $T:V \rightarrow V$ then the characteristic equation of $T$, $X_T(x)$ has the property that $X_T(A)=0$ where A is the matrix representation of the transformation. Equivalently $X_A(A)=0$?

Can anyone explain to me why this is equivalent to $m_T|X_T$ where $m_T$ is the minimum polynomial of $T$?

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    Dear LHS: $I$ suggest that you answer your own question. --- If you write a comment for me (or for anybody), $p$lease try to think of using the `@`sign.2012-01-04

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There's little to add to the comment by Pierre-Yves Gaillard here... Denoting by $K$ the ground field of vector space $V$, we observe that the set of all polynomials $p\in K[x]$ such that $p(T)=0$ is an ideal. Since $K[x]$ is a principal ideal domain, it follows that there exists $m_T\in K[x]$ (unique up to a unit in $K[x]$, that is a nonzero scalar) such that $\{p\in K[x]:p(T)=0\}=\{p\in K[x]: m_T\text{ divides }p\}$ Which was to be explained.