Timeline for Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity
Current License: CC BY-SA 3.0
13 events
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| Oct 11, 2013 at 2:09 | comment | added | Suvrit | one minor point worth noting is that if $[\alpha_i^{j-1}\beta_j^{i-1}]$ is rank-1, (I'm running indices from $1,2,\ldots$), then $M=\text{diag}(\alpha)Z\text{diag}(\beta)$, which is clearly nonsingular. This begs the question: what is the largest rank that the matrix formed from the $\alpha$s and $\beta$s can have? | |
| Oct 10, 2013 at 19:22 | comment | added | Peter Mueller | Checking all possibilities for $\alpha$ and $\beta$, the answer is affirmative for all $\ell\le12$. | |
| Oct 10, 2013 at 18:09 | comment | added | Suvrit | @Peter: I used $\alpha$ a random $\pm 1$ vector, and $\beta=[-1,1,-1,\ldots,]$. But I think I'm just running into round-off--- | |
| Oct 9, 2013 at 14:51 | comment | added | Binzhou Xia | @PeterMueller: Yes. It seems that the fewer prime factors $l$ has, the easier the situation would be. So I think one may start the question when $l=p$ is prime (the $2p$th cyclotomic polynomial is also clear). | |
| Oct 9, 2013 at 14:23 | comment | added | Peter Mueller | This question has the flavor of a theorem of Chebotarev on the non-singularity of the minors of the matrix with entry $\zeta^{ij}$ in position $(i,j)$, where $\zeta$ is a primitive $p$-th root of unity for a prime $p$. | |
| Oct 9, 2013 at 14:20 | comment | added | Peter Mueller | @suv...rit: Which kind of reformulation did you use? For $l=40$, there are $2^{80}$ possibilities for the vectors $\alpha$ and $\beta$ ... | |
| Oct 9, 2013 at 14:08 | comment | added | Suvrit | for me think broke at $l=40$, but should also break earlier---but I still suspect that this might be due to numerical roundoff (i.e., numerical rank was $l-1$) | |
| Oct 9, 2013 at 8:03 | comment | added | Binzhou Xia | @suv....rit :Thanks! In your experiments, what is the smallest value of $l$ such that the matrix seems to be singular? | |
| Oct 8, 2013 at 18:53 | comment | added | Suvrit | My experiments suggest that this matrix can be singular---however, due to numerical concerns, am not yet conclusively claiming singularity. | |
| Oct 3, 2013 at 5:42 | history | edited | Binzhou Xia | CC BY-SA 3.0 |
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| Oct 2, 2013 at 16:40 | history | edited | Binzhou Xia |
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| Oct 2, 2013 at 16:31 | history | edited | Binzhou Xia | CC BY-SA 3.0 |
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| Oct 2, 2013 at 7:49 | history | asked | Binzhou Xia | CC BY-SA 3.0 |