Timeline for How to efficiently evaluate the action form of a matrix power?
Current License: CC BY-SA 3.0
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| May 17, 2017 at 10:39 | history | edited | J. M. isn't a mathematician |
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| May 15, 2017 at 13:55 | comment | added | Henry.L | @J.M.isn'tamathematician You are right...I am missing a Jordan block case...see this post mathoverflow.net/questions/23629/… | |
| May 15, 2017 at 13:48 | comment | added | J. M. isn't a mathematician | @Henry, $\begin{pmatrix}1 & 1 & 0 \\0 & 1 & 1 \\0 & 0 & 1\end{pmatrix}$ and larger versions of it were what I had in mind for "not diagonalizable" (i.e. defective). | |
| May 15, 2017 at 13:45 | comment | added | Henry.L | And therefore SVD will definitely save some computational cost if we only need to store $B$ and a diagonal $\Lambda$...Am I missing anything now? | |
| May 15, 2017 at 13:42 | comment | added | Henry.L | @J.M.isn'tamathematician I used singular/eigenvalue decomposition for the same thing, you can tell from my notations....they are all diagonalizable since the order is finite and they are on $\mathbb{C}$ which is algebraically closed...If you read OP carefully, you will see the only obstacle is that the matrix is too big... | |
| May 15, 2017 at 12:48 | comment | added | J. M. isn't a mathematician | @Henry, that doesn't look like an SVD to me, and I'm also missing on how one might use SVD to efficiently evaluate a matrix function, much less the action form. If you intended to present the eigendecomposition, then you of course know that not all matrices are diagonalizable. | |
| May 15, 2017 at 10:36 | comment | added | Henry.L | You cannot adapt this technique. But another common technique is to perform a singular decomposition $A=B^T\Lambda B$ where $\Lambda$ is diagonal. If such a diagonalization presents special structure, then you can probably hope that you do not need to write all steps down. | |
| May 15, 2017 at 10:30 | comment | added | Federico Poloni | You can't. As far as I know, everyone does the products one after the other ($v_{k+1}=Av_k$), in practice. This is a very common operation (e.g., Arnoldi method, power iteration...), so it would be very strange if someone came up with a better method all of a sudden. | |
| May 15, 2017 at 10:27 | review | First posts | |||
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| May 15, 2017 at 10:17 | history | asked | マダオ | CC BY-SA 3.0 |