uniqueness of Jordan form

Existence says every complex matrix HAS a Jordan form. Uniqueness says it has essentially only ONE: the list of Jordan blocks — which eigenvalues appear and with which block sizes — is completely determined by A. The only freedom is the order in which you arrange the blocks along the diagonal.

Precisely: if P^-1 A P = J and Q^-1 A Q = J' are both Jordan forms of A, then J and J' have the same multiset of blocks, so J' is obtained from J by permuting blocks (and the eigenvectors/chains can differ). The proof is the rank-jump formula: for each lambda the numbers dim ker(A - lambda*I)^k are invariant under similarity, and they pin down every block size, leaving no choices.

This is what makes the Jordan form a true canonical form: it is a representative of the similarity class that is unique up to a trivial relabeling. Pick a fixed ordering convention (say, by eigenvalue then by decreasing block size) and the form becomes literally unique — a single normal form per similarity class.

The payoff is the classification theorem: two complex matrices are similar if and only if they have the same Jordan form. Uniqueness is the 'only if' direction made sharp — same operator, same blocks, no ambiguity.