Jordan chain

A Jordan chain is the sequence of vectors that produces one Jordan block. Start from an eigenvector and keep solving 'one level up': you build a ladder v_1, v_2, ..., v_k where the operator pushes each rung down to the previous one (plus a copy of lambda times itself). These chained vectors are exactly the right basis to expose a single block.

Precisely, for A with eigenvalue lambda, a Jordan chain of length k is vectors v_1, ..., v_k with (A - lambda*I) v_1 = 0 and (A - lambda*I) v_j = v_{j-1} for j = 2, ..., k. The bottom vector v_1 is an honest eigenvector; the others are generalized eigenvectors, since (A - lambda*I)^j v_j = 0 but (A - lambda*I)^{j-1} v_j != 0. The vector v_k that starts the chain is called the lead (or top) generalized eigenvector.

On the span of one chain, in the ordered basis v_1, ..., v_k, the operator acts exactly as a single Jordan block J_k(lambda): A v_1 = lambda v_1 and A v_j = lambda v_j + v_{j-1}. So a chain and a block are two views of the same object — the block is the matrix, the chain is the basis.

Practically, you find chains by working DOWN from the top: pick a vector in ker(A - lambda*I)^k that is not killed by the lower power, then apply (A - lambda*I) repeatedly to read off the rest of the ladder. Independent chains, stacked, give the full Jordan basis.