index of nilpotency
An operator N is nilpotent if some power of it is zero. The index of nilpotency is the SMALLEST positive integer k for which N^k = 0 (while N^(k-1) is still nonzero). It measures how many times you must apply N before everything collapses to zero — the depth of the nilpotency.
The index ties directly to the minimal polynomial. For a nilpotent operator the only eigenvalue is 0, the minimal polynomial is exactly x^k, and its degree k is the index. So 'find the index' and 'find the minimal polynomial of a nilpotent' are the same task. For a general operator, the index of an eigenvalue lambda is the smallest k with (T - lambda I)^k vanishing on the generalized eigenspace — i.e. the exponent of (x - lambda) in the minimal polynomial.
Geometrically the index is the size of the LARGEST Jordan block for that eigenvalue. A nilpotent operator built from Jordan blocks of sizes 4, 2, 1 has index 4: the longest chain needs four applications to die. This is why the minimal polynomial records largest-block sizes while the characteristic polynomial records total sizes — the index is the bridge between the algebra and the block geometry.
Practically the index controls a lot. It bounds how high you must go when computing generalized eigenspaces (stop at the index), it equals the length of the longest Jordan chain, and it is exactly the threshold where the kernels ker(T - lambda I)^j stop growing. Knowing it tells you the worst-case 'shear depth' of the operator at lambda.
A size-3 nilpotent block has index 3: it takes three applications to reach zero, and the minimal polynomial is x^3.
The index is at most the algebraic multiplicity (largest block <= total size) and at most dim V. An operator is diagonalizable at lambda exactly when the index there is 1 — every Jordan block at lambda has size one.