diagonalizable as a special Jordan form

Diagonalization is the happy case you met in a first course: A = P D P^-1 with D diagonal, when there is a full eigenbasis. Jordan form is the universal upgrade: every operator over C has one, and diagonalizable is simply the case where every Jordan block has size 1 — no superdiagonal 1s at all.

The dividing line is the gap between algebraic and geometric multiplicity. A is diagonalizable exactly when, for every eigenvalue, geometric multiplicity equals algebraic multiplicity — equivalently when the minimal polynomial has no repeated roots. Any eigenvalue short on eigenvectors (a defective eigenvalue) forces a block of size >= 2 and a genuinely non-diagonal Jordan form.

So Jordan form generalizes diagonalization in the cleanest possible way: it keeps everything diagonalization gives (a basis adapted to the operator, easy powers, easy functions) and adds exactly the minimal extra structure — Jordan chains, off-diagonal 1s — needed to cover the defective operators a single eigenbasis cannot reach.

Practically this reframes 'is A diagonalizable?' as 'are all Jordan blocks size 1?', a question the rank-jump formula answers from kernel dimensions. And it explains why diagonalizable matrices are 'generic': a random matrix has distinct eigenvalues with probability 1, hence all size-1 blocks; defectiveness is the measure-zero exceptional case Jordan form was invented to handle.