<s>[INST] Jordan basis unique? Is Jordan basis unique? I have a 4x4 matrix, find eigenvectors and one generalized eigenvector, also trying different linearly independent eigenvectors but the matrix P so that PJP^-1 =A only works for certain vector.
\begin{pmatrix}-\frac{1}{2}&0&-\frac{1}{2}&0\\ \frac{3}{2}&0&\frac{1}{2}&0\\ -\frac{3}{2}&0&-\frac{3}{2}&-1\\ 1&1&1&1\end{pmatrix}\begin{pmatrix}1&-1&0&-1\\ 0&2&0&1\\ -2&1&-1&1\\ 2&-1&2&0\end{pmatrix}\begin{pmatrix}1&1&0&0\\ -1&0&1&1\\ -3&-1&0&0\\ 3&0&-1&0\end{pmatrix}=\begin{pmatrix}-1&0&0&0\\ 0&1&0&-1\\ 0&0&1&1\\ 0&0&0&1\end{pmatrix}
 [/INST] It is not: the Jordan matrix is unique, but there are several basis which will yield that matrix.
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