= Jordan chain
{c}
{title2=$v_1,\ldots,v_r$}
A Jordan chain of a <linear operator> $T$ for an <eigenvalue> $\lambda$ is a list of nonzero vectors satisfying $(T-\lambda I)v_1=0$ and $(T-\lambda I)v_j=v_{j-1}$ for $2\leq j\leq r$. These vectors are <linearly independent>: in a vanishing <linear combination>, applying the largest relevant power of $T-\lambda I$ isolates the last coefficient times $v_1$, then descending induction removes all coefficients. On their span the operator has one <Jordan block>. This is different from a <generalized eigenfunction> that solves $(T-\lambda I)u=0$ outside the original function space.
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