Timeline for Bounding the non-multiplicativity of isometric projection
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 22, 2016 at 18:48 | vote | accept | Asaf Shachar | ||
| Sep 17, 2016 at 17:32 | history | bounty ended | Asaf Shachar | ||
| Sep 17, 2016 at 15:49 | history | edited | fedja | CC BY-SA 3.0 |
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| Sep 17, 2016 at 15:27 | history | edited | fedja | CC BY-SA 3.0 |
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| Sep 16, 2016 at 16:30 | comment | added | fedja | $U$ is unitary, so all eigenvalues of $U$ are on the unit circle. If one of them is far from $1$, it is also far from the positive semi-axis. Also $\|U-I\|$ (the operator norm) is just $\max_{\lambda}|\lambda-1|$ | |
| Sep 16, 2016 at 16:19 | comment | added | Asaf Shachar | Thanks. Unfortunately some things are still not clear to me. I tried to write a more detailed version of your answer as I understand it (CW). I am stuck already at the point of showing existence of such an eigenvalue of $U$, which is far enough from the $x$-axis. (Please see my precise difficulty in the "Attempted proof of the lemma"). | |
| Sep 16, 2016 at 13:29 | comment | added | fedja | 1) Yes. Actually $U=V$. I was just interrupted when writing and changed the letter for no apparent reason when came back 2) Equal to $\delta$. You are always fine with any $\ell^p$ operator norm, the "row sum" rule arises when $p=\infty$ but I prefer to deal with $\ell^2$ here. 3) $x$-axis, of course. Any connected component of the union of Gershgorin disks contains as many eigenvalues as the number of disks in it and you cannot build a connected chain of length $C\delta/2$ out of $n$ disks of radius $\delta$ if $C>5n$. | |
| Sep 16, 2016 at 13:08 | comment | added | Asaf Shachar | Thanks! However, I am not sure I follow all the details. Would you mind clarifying the following points: (1) In your answer, is U itself a unitary operator? (2) How can you choose the radius of the Gershgorin disks "at your will"? They are determined by the sum of the non-diagonal terms in each row. (I guess you have used implicitly some estimate you did not state). (3) How did you deduce the existence of an eigenvalue far from the positive semi-axis? (I am also not sure what do you mean by "positive semi-axis"; Is it half of the x-axis or the y-axis)? Thanks for your patience. | |
| Sep 15, 2016 at 22:03 | history | answered | fedja | CC BY-SA 3.0 |