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I can easily compute shur's decomposition but I need Jordan normal form: A = PJP_inv, if I have shur's decomposition in hand is there a method of recovering P if I know the eigenvalues of A?

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  • $\begingroup$ Just use the normal method. Because the matrix is upper triangular, it’s easy to find the generalized eigenvectors. $\endgroup$ – Omnomnomnom May 22 at 22:48
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    $\begingroup$ $\newcommand{\P}{\widetilde{P}}$As Omnomnomnom says, if you have $A = QUQ^{-1}$ (Schur decomposition of $A$), find the Jordan normal form of $U$ first using your standard approaches (and since $U$ is upper triangular, it should be easier to do than for a general matrix); you'll get say $U = \P J \P^{-1}$. Then $$A = QUQ^{-1} = Q \P J \P ^{-1} Q^{-1},$$ which is in Jordan normal form $PJP^{-1}$ with $P = Q\P$. $\endgroup$ – Minus One-Twelfth May 22 at 22:58

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