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Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af be the null space of Af.

Then ( A ) Af ; ;

( B ) each Af is invariant under T ; ;

( C ) if Af is the operator induced on Af by T, then the minimal polynomial for Af is Af.

Proof.