Timeline for Show that a certain ratio of diagonal entries dominates a certain ratio of singular values
Current License: CC BY-SA 3.0
10 events
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| Apr 9, 2017 at 21:13 | comment | added | Paul B. Slater | Well, it seems that I should better inform myself as to the scope of the different stack exchanges. I can understand the desire for definite boundaries, but many areas are "interdisciplinary". Are the stack exchanges designed to correspond to specific disciplines? I'll definitely be more circumspect about cross-posting in the future--since it seems to annoy folks. | |
| Apr 9, 2017 at 13:17 | comment | added | Jyrki Lahtonen | Please comment on whether this question fits, in your opinion, MO better than MSE. My guess would be that this may even be off-topic at MO. | |
| Apr 7, 2017 at 21:39 | comment | added | D.W. | Cross-posted: mathoverflow.net/q/262943/37212, math.stackexchange.com/q/2157393/14578. Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. | |
| Apr 2, 2017 at 0:42 | comment | added | Paul B. Slater | To add to the interesting observation of VorKir, "the inequality turns into the equality", not only in the diagonal case, but more broadly, if only the (off-diagonal) 12-, 21-, 34- and 43-entries are zero. | |
| Mar 1, 2017 at 18:05 | comment | added | Paul B. Slater | OK, let me try to get this right using the notation of the problem I put (rather than the one I had been working with). Now, $q=\exp \left(-\cosh ^{-1}\left(\frac{\frac{2 d_{12} d_{34} p}{\sqrt{d_{11} d_{22}} \sqrt{d_{33} d_{44}}}-p^2-1}{2 \sqrt{\frac{d_{12}^2}{d_{11} d_{22}}-1} \sqrt{\frac{d_{34}^2}{d_{33} d_{44}}-1} p}\right)\right)$. | |
| Mar 1, 2017 at 2:11 | comment | added | Paul B. Slater | My apologies--I mixed some notation. Actually, $\mu \equiv p$. See Lemma 5 in arxiv.org/pdf/1610.01410.pdf for what I used to get the formula. In that paper, $\epsilon$ is the ratio of singular values, denoted $q$ in my original question. (I didn't fully understand your earlier point about a similarity transform.) | |
| Mar 1, 2017 at 0:09 | comment | added | VorKir | What is $\mu$ in your formula? In general, it would be interesting to know the background, just out of curiosity. I think there should be a nice and elegant argument working to proof the inequality, but cannot suggest it:) | |
| Feb 28, 2017 at 23:57 | comment | added | Paul B. Slater | I verified the (equality) assertion of VorKir in the diagonal case. Further, "the behavior of the singular values ratio as a function of two variables" that he requests is $\epsilon=\exp \left(-\cosh ^{-1}\left(\frac{2 d_{12} d_{34} \mu -\mu ^2-1}{2 \sqrt{d_{12}^2-1} \sqrt{d_{34}^2-1} \mu }\right)\right)$. I can give some interesting background (shortly to be an arXiv posting) on this result, if requested. | |
| Feb 27, 2017 at 19:13 | comment | added | VorKir | As a brute force, one can show that in diagonal case the inequality turns into the equality, and introducing two parameters for $d_{12}$ and $d_{34}$ just analyze the behavior of singular values ratio as a function of two variables. It may be more convenient then to do a similarity transform to $D_1^{-1/2} D_2 D_1^{-1/2}$ and work with $D_1^{-1} D_2$ | |
| Feb 23, 2017 at 15:05 | history | asked | Paul B. Slater | CC BY-SA 3.0 |