generalized eigenvector
An ordinary eigenvector v satisfies (T - lambda I)v = 0 — one hit by T - lambda I kills it. A generalized eigenvector is more forgiving: some power of T - lambda I kills it. It may survive one or two hits, but eventually it falls to zero.
Precisely, v != 0 is a generalized eigenvector for lambda if (T - lambda I)^k v = 0 for some positive integer k. The set of all such v (together with 0) is the generalized eigenspace ker( (T - lambda I)^m ), where m is the algebraic multiplicity. Unlike ordinary eigenspaces, these generalized eigenspaces always fill up V when the characteristic polynomial splits — their dimensions are the full algebraic multiplicities, not the smaller geometric ones.
Generalized eigenvectors organize into Jordan chains: a string v1, v2, ..., vj where (T - lambda I) sends each vector to the previous one and v1 down to zero. Each chain is a cyclic subspace and produces one Jordan block. This is why Jordan form exists at all — there are not enough genuine eigenvectors, so you complete the basis with generalized ones.
[0; 1] is not an eigenvector but is killed by (T - 5I)^2, so it is a generalized eigenvector forming a length-2 Jordan chain with [1; 0].
On a generalized eigenspace the operator N = T - lambda I is nilpotent; that nilpotent restriction is what carries all the non-diagonalizable structure.