In Jordan decomposition of a complex square matrix $M = P J P^{-1}$, the Jordan canonical form $J$ is unique up to permutation of the diagonal Jordan blocks $J_i$'s along the diagonal. $P$ consists of a generalized eigenbasis as its columns.
If a complex or real square matrix can be similar to a diagonal matrix $M = P D P^{-1}$, the diagonal matrix $D$ is unique up to permutation of the diagonal entries along the diagonal line. $P$ consists of an eigenbasis as its columns.
- If a complex or real square matrix can be unitarily/orthogonally similar to a diagonal matrix $M = P D P^{H}$, the diagonal matrix $D$ is unique up to permutation of the diagonal entries along the diagonal line. $P$ consists of an unitarynormal/orthonormal eigenbasis as its columns.
I wonder what we can say about the uniqueness of $P$ in each case? Can it be unique up to some matrix transformation?
Thanks!