Timeline for SVD of $2\times 2$ matrix where entries have different units
Current License: CC BY-SA 4.0
9 events
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| May 6, 2021 at 22:27 | history | edited | Cosmas Zachos | CC BY-SA 4.0 |
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| May 6, 2021 at 22:05 | comment | added | Cosmas Zachos | I'm sorry, The Pauli vector method is also overkill. Will write a simpler answer. Note N can be transformed to a symmetric, hence orthogonally diagonalizable matrix by a further similarity transformation... | |
| May 6, 2021 at 21:52 | comment | added | a_guest | It appears I was too fixated on arriving at that $\exp$ term, so I got the order wrong. Of course it should be $N^n = \exp n\log N$. But here I don't see how this can be converted to the form $\exp ia(\hat{z}\vec{\sigma})$. How do I determine $a$ in this case? | |
| May 6, 2021 at 21:34 | comment | added | a_guest | So in the end I could express $M^n$ as a function of $\mathbb{1}$ and $\hat{z}\vec{\sigma}$ by using the binomial theorem? In order to use Euler's formula, I would start by writing $M^n = \log\exp nM$ and then transform this expression until I arrive at the form $\exp ia(\hat{z}\vec{\sigma})$ (with $a = -inz$). Is that correct? | |
| May 6, 2021 at 21:26 | vote | accept | a_guest | ||
| May 6, 2021 at 15:49 | comment | added | Brick | As an add-on: This procedure could be viewed as a specific case of normalizing $M$, which is useful and, if you're computing numerically, sometimes necessary for the results to make sense or to be numerically stable. Even if you didn't have the problem with units, if the singular values are very different in magnitude you might still have problems that could be alleviated by a modified application of this procedure. This is especially true if your matrix is related to analysis of random variables, e.g. stats.stackexchange.com/questions/12200/… | |
| May 6, 2021 at 15:33 | history | edited | Cosmas Zachos | CC BY-SA 4.0 |
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| May 6, 2021 at 13:36 | history | edited | Cosmas Zachos | CC BY-SA 4.0 |
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| May 6, 2021 at 13:27 | history | answered | Cosmas Zachos | CC BY-SA 4.0 |