Timeline for A simple but curious determinantal inequality
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 26, 2017 at 8:54 | vote | accept | Wolfgang | ||
| Jun 25, 2017 at 0:00 | comment | added | M. Lin | Sorry to have confused you, I had known this $\det (A^2+BAAB)\geqslant \det(A^2+ABBA)$. It follows from COROLLARY 2.3 of the paper M. Lin, H. Wolkowicz, An eigenvalue majorization inequality for positive semidefinite block matrices, Linear Multilinear Algebra 60 (2012), no. 11-12, 1365–1368. tandfonline.com/doi/abs/10.1080/03081087.2011.651723 | |
| Jun 24, 2017 at 13:41 | answer | added | jjcale | timeline score: 3 | |
| Jun 24, 2017 at 2:38 | answer | added | Suvrit | timeline score: 4 | |
| Jun 23, 2017 at 20:46 | comment | added | Wolfgang | @Suvrit of course, I know. But I found it funny to notice that this means the two terms of the sum can have any difference of "magnitude". :) | |
| Jun 23, 2017 at 20:40 | comment | added | Suvrit | The multiplication by $\lambda$ is a red-herring, because you could trivially replace $B$ by $\sqrt{\lambda}B$ without changing anything, so we might as well prove the $\lambda=1$ case. | |
| Jun 23, 2017 at 13:25 | comment | added | Wolfgang | Just noted that it can be written more symmetrically as follows: Putting $C:=A^{-1}BA$ and $D:=ABA^{-1}$, then $\boxed{\det(A^k+CD)\geqslant \det(A^k+DC)}$. | |
| Jun 23, 2017 at 12:12 | comment | added | Wolfgang | @FedorPetrov That looks like an appealing idea, but I already have a hard time following how he re-arranges the traces. And no idea how to use that method with inverses, let alone arbitrary powers, even the k=2 case. | |
| Jun 23, 2017 at 11:45 | comment | added | Fedor Petrov | have you tried Tao's strategy: proving that the spectrum of $C=A^{-(k+1)/2}BABA^{-(k+1)/2}$ is majorated by the spectrum of $D=A^{-(k-1)/2}BA^{-1}BA^{-(k-1)/2}$? | |
| Jun 23, 2017 at 11:14 | comment | added | Wolfgang | @Suvrit Yes I wouldn't have posted that without checking first for several thousands of random matrices of different sizes and several big and small values of $k$. I almost thought my few lines of PARI code had a bug... :-) | |
| Jun 23, 2017 at 11:06 | comment | added | Suvrit | I have some other generalizations based on your very interesting determinantal conjectures --- I wish I had time to think about these. Do you have numerical evidence for these? The second one suggests that perhaps there is even a Löwner dominance somewhere that yields the ineq for all $\lambda\ge 0$. | |
| Jun 23, 2017 at 10:35 | history | edited | Wolfgang | CC BY-SA 3.0 |
introduced exponent $k$ :)
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| Jun 23, 2017 at 10:16 | history | asked | Wolfgang | CC BY-SA 3.0 |