Timeline for Finding Toeplitz matrix nearest to a given matrix
Current License: CC BY-SA 3.0
20 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Mar 15, 2018 at 21:45 | comment | added | Rodrigo de Azevedo | @Creator I am not surprised. My first encounter with Toeplitz matrices was via digital signal processing. I studied the MUSIC algorithm in a "past life", too, but remember nothing, sadly. Thank you for the information. | |
| Mar 15, 2018 at 21:43 | comment | added | Creator | @RodrigodeAzevedo Yes, it is purely Engineering application. It is related to the MUSIC algorithm. If your google MUSIC algorithm you should get the idea. It relates to target direction in radar. | |
| Mar 15, 2018 at 20:50 | comment | added | Rodrigo de Azevedo | @Creator If I may ask, where did this optimization problem come from? What is its background? Did it have an engineering application, for instance? | |
| Mar 15, 2018 at 20:38 | comment | added | Creator | @RodrigodeAzevedo The directions of the Eigenvectors of the such Toeplitz matrix deviates considerably from Eigenvectors of A when the condition number of A is high. I see the comment from Federico regarding Eigenvectors. In any case, is there any other way to get a Toeplitz or put constraint on Eigenvectors? | |
| Mar 15, 2018 at 20:28 | vote | accept | Creator | ||
| Mar 15, 2018 at 20:27 | comment | added | Creator | @RodrigodeAzevedo Yes, in my first read, I did not understand. Now, I am clear. I have implemented the Toeplitz matrix in this way (your way) few years back for my work, but never knew the method was optimal. Therefore, I was looking for an alternative way to get a Toeplitz matrix. In any case, I am happy to see that the solution can be derived as a minimal norm, so full credit to you. Thanks. | |
| Mar 15, 2018 at 19:51 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Added ratios of inner products
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| Mar 15, 2018 at 15:12 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Fixed silly typo, minor edit
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| Mar 15, 2018 at 14:27 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Added section on the Toeplitz matrix nearest to the given symmetric matrix
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| Mar 15, 2018 at 14:14 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Minor edits
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| Mar 15, 2018 at 14:08 | comment | added | Rodrigo de Azevedo | @FedericoPoloni I used your suggestion and considered also the case where the squared Frobenius norm is minimized. The solution is indeed trivial. Updated my answer. | |
| Mar 15, 2018 at 14:07 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Appended new section on the Frobenius norm
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| Mar 15, 2018 at 11:29 | comment | added | Rodrigo de Azevedo | @FedericoPoloni Indeed. I didn't know how to answer the original question, so I took the liberty of minimizing the spectral norm of the difference instead. | |
| Mar 15, 2018 at 10:58 | comment | added | Federico Poloni | Two matrices that are close in the Euclidean norm do not always have close eigenvector matrices (even after one takes care that these objects are uniquely defined --- see my other comment for this). For instance, $I+\varepsilon A$ and $I+\varepsilon B$ are very close matrices (in the Euclidean norm) that have the same sets of eigenvectors as $A$ and $B$, respectively, for arbitrary $A$ and $B$. | |
| Mar 15, 2018 at 10:06 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
Fixed typo, minor edit
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| Mar 15, 2018 at 9:30 | comment | added | Rodrigo de Azevedo | I don't think you understood my approach. Note that, for $n=3$, $$\begin{bmatrix} x_1 & x_2 & x_3\\ x_2 & x_1 & x_2\\ x_3 & x_2 & x_1\end{bmatrix} = x_1 \begin{bmatrix} 1 & 0 & 0\\ 0 & 1& 0\\ 0 & 0 & 1\end{bmatrix} + x_2 \begin{bmatrix} 0 & 1 & 0\\ 1 & 0& 1\\ 0 & 1 & 0\end{bmatrix} + x_3 \begin{bmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 0\end{bmatrix}$$ | |
| Mar 15, 2018 at 0:33 | comment | added | Creator | The question was related to an arbitrary matrix not a sequence of matrix. | |
| Mar 14, 2018 at 21:47 | history | edited | Rodrigo de Azevedo | CC BY-SA 3.0 |
added 4 characters in body
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| Mar 14, 2018 at 21:39 | history | answered | Rodrigo de Azevedo | CC BY-SA 3.0 |