= Cayley-Hamilton theorem from cyclic subspaces
{c}
{title2=$\chi_T(T)=0$}
A <cyclic vector> gives a <companion matrix>, whose characteristic relation annihilates its generator and hence every iterate. If there is no cyclic vector spanning the whole space, a nonzero proper <cyclic subspace> is invariant. Induction annihilates the restriction and the quotient; the quotient polynomial first sends the whole space into the invariant subspace, and the restriction polynomial then kills it. The block-triangular <characteristic polynomial> is the product of these two polynomials.
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