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As it turns out, the author does cover the rank-decomposition (under the name "Rank-Revealing Factorization") in his blog as well:

https://nhigham.com/2021/05/19/what-is-a-rank-revealing-fact...

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Higham there adds the condition that X, Y be well-conditioned (as they are in the SVD, for example). That's a problem with the Jordan decomposition A = X J X^-1: the X is non-singular in theory, but can be ill-conditioned, and thus the Jordan decomposition is not stable and not really used in practice (as highlighted in the original article).

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