Cayley-Hamilton theorem

Theorem. (Cayley-Hamilton) Let T be a k-linear endomorphism of a finite-dimensional vector space V over a field k. Let P_T(x) be the characteristic polynomial

\displaystyle P_T(x)=\det(x\cdot 1_V-T)

Then

\displaystyle P_T(T)=0\in\mathrm{End}_k(V).

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