elementary divisors
If the invariant factors are the coarse fingerprint, the elementary divisors are the same information smashed down to its prime atoms. Take each invariant factor, factor it into powers of irreducible (prime) polynomials, and collect all those prime-power pieces together with multiplicity: that bag of prime powers is the list of elementary divisors.
Formally: factor each f_i over F into irreducibles, f_i = p1^(e1) p2^(e2) ..., and the elementary divisors are all the individual factors p^e that appear, listed across all invariant factors. Because the f_i form a divisibility chain, you can reconstruct the invariant factors back from the elementary divisors uniquely — the two lists carry exactly the same content, just reorganized.
Their headline role: over an algebraically closed field every irreducible is linear, p = x - lambda, so each elementary divisor is (x - lambda)^e — and that is precisely one Jordan block of size e for eigenvalue lambda. So the elementary divisors ARE the Jordan blocks. Over a general field, where some irreducibles have higher degree, each elementary divisor instead gives a single companion (or primary) block.
Heads-up: elementary divisors depend on the field, because what counts as irreducible depends on the field. Over the reals x^2 + 1 is prime; over the complexes it splits as (x-i)(x+i). So the elementary divisors can refine when you extend the field, even though the invariant factors do not.
Group prime powers: largest of each prime goes into f2, the leftover (x-2) into f1. Over C these three elementary divisors are three Jordan blocks.
To recover invariant factors from elementary divisors: for each prime, line up its powers; the largest powers across all primes multiply to fk, the next-largest to f(k-1), and so on down the chain.