0
$\begingroup$
  1. 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.

  2. 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.

  3. 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!

1 Answers 1

1

Note that, in the three cases, if $M=I$ then $D=I$ and $P$ can be any invertible matrix. So you cannot expect much uniqueness in general.

  • 0
    It still depends a lot. Take $D=I$, and still $P$ can be any invertible matrix. Take $D$ diagonal with all diagonal entries different, and then if you fix order, $P$ is unique. Take $J$ to be a full Jordan block, and you can still multiply $P$ by any invertible upper-triangular Toeplitz matrix.2012-11-25