min/char poly from invariant factors

The invariant factors are not just a classification gadget — they let you read the minimal and characteristic polynomials straight off, with almost no work. Once you have the chain f1 | f2 | ... | fk, two of an operator's most important polynomials are sitting right there.

The rule is clean. The characteristic polynomial is the product of all the invariant factors: charpoly = f1 f2 ... fk (its degree is therefore n, the dimension, since the degrees add up). The minimal polynomial is the single largest invariant factor: minpoly = fk. That fk is divisible by all the others is exactly why it annihilates every cyclic piece, hence all of V.

This makes several classic facts transparent. The minimal polynomial divides the characteristic polynomial (fk divides the product) — that is Cayley-Hamilton, read off instantly. Minpoly and charpoly have the same irreducible factors (each prime in fk also appears in the product), explaining why they share the same roots, namely the eigenvalues. And the operator is cyclic (nonderogatory) exactly when there is one invariant factor, i.e. minpoly = charpoly.

A subtle point worth stating: the minimal and characteristic polynomials alone do NOT determine the similarity class in general — they only pin down the largest invariant factor and the total degree. You need the full chain f1, ..., fk to know everything. Two non-similar operators can share both minpoly and charpoly while differing in the middle factors.