rational form vs Jordan form

Rational canonical form and Jordan canonical form are two outputs of the same machine, differing only in which currency they spend. RCF is built from invariant factors and companion blocks and works over any field; JCF is built from elementary divisors and lives only over a field where the characteristic polynomial fully factors (an algebraically closed field, or at least a splitting field).

Trace it through the module picture. The structure theorem gives V two descriptions: invariant-factor form yields the companion blocks of RCF, while elementary-divisor form yields, over a splitting field, blocks (x - lambda)^e. Each such block is one Jordan block of size e at eigenvalue lambda. So elementary divisors and Jordan blocks are literally the same data; JCF is RCF reorganized around eigenvalues once they exist.

The practical trade-off: JCF is more transparent when you have eigenvalues — you read off lambda's directly, and powers, exponentials, and functions of the matrix are easy on near-diagonal blocks. RCF is more robust — it never leaves the base field, never needs roots, and is therefore the right tool over Q, finite fields, or whenever you refuse to extend.

A clean way to hold both: over C the two forms carry identical information and you can convert freely. Over R or Q, JCF may simply not exist (think of a rotation with no real eigenvalues) while RCF always does. When a real matrix has complex eigenvalues, its RCF uses a degree-2 companion block where JCF would have needed a complex 2-by-2 Jordan block.