Timeline for Is it possible to prove the Jordan decomposition starting from Schur's decomposition?
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| Jul 13, 2020 at 9:13 | comment | added | Federico Poloni | Another comment: the way I see it, there are two "ingredients" needed in the proof: (1) write the matrix as a direct sum (block diagonal matrix) of upper triangular matrices, each one with a different eigenvalue $\lambda_i$ on the diagonal, and (2) transform those diagonal blocks into Jordan form. I don't think you considered (1) here, but that is a tricky step, because the transformations that you need must involve explicitly $\frac{1}{\lambda_i - \lambda_j}$ to clear the off-diagonal block $(i,j)$ (otherwise they would work also if the $\lambda_i = \lambda_j$, which is not the case). | |
| Jul 13, 2020 at 9:09 | comment | added | Federico Poloni | every upper triangular matrix with zero diagonal entries can be brought to every possible Jordon block under a suitable basis No, there are some rank constraints. You can't transform $$\begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{bmatrix}$$ into You can't transform $$\begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{bmatrix}$$, for instance. | |
| Jul 13, 2020 at 9:06 | vote | accept | Fdost | ||
| Jul 13, 2020 at 9:05 | comment | added | Fdost | Thanks for the wonderful explanation and the examples. I guess I understand it better now. Particularly, if the Jordan form respects the structure of D, then commutativity follows since each block would correspond to a scaled identity block in D and commutativity is obvious. Is the following observation true? As one can start with an arbitrarily structured D and add an arbitrary upper triangular matrix with zero diagonal entries, does this mean then, every upper triangular matrix with zero diagonal entries can be brought to every possible Jordon block form under a suitable basis? | |
| Jul 13, 2020 at 7:25 | history | edited | Federico Poloni | CC BY-SA 4.0 |
added 8 characters in body
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| Jul 13, 2020 at 7:20 | history | answered | Federico Poloni | CC BY-SA 4.0 |