Timeline for A metric for Grassmannians
Current License: CC BY-SA 2.5
20 events
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| Aug 1, 2020 at 15:59 | comment | added | Pedro G. Mattos | I also wanted to know if there is some way to define an "angular" metric, as can be done with angle between lines if the Grassmannian is the projective space, i.e. Grassmannian for k=1. Is there a way to define angles between planes, 3-spaces etc? | |
| Aug 1, 2020 at 15:57 | comment | added | Pedro G. Mattos | "This is I think not quite the same as the one suggested by Ryan Budney". Is it because in your definition the considers the infimum over all v, not only those with unit norm? | |
| Mar 18, 2015 at 20:56 | comment | added | Jairo Bochi | @IanMorris Let me consider a variation of one of Ryan's suggestions: If we metrize the projective space using the sine of the angle, and regard the planes V, W as subsets of the projective space, then the Hausdorff distance between them is exactly Ian's distance. Using this interpretation, Ian's discovery that the two maxima in his formula coincide become a corollary of the fact that there exists an isometry of the ambient space that switches V and W (see math.stackexchange.com/questions/118873/…). | |
| Feb 9, 2014 at 14:18 | comment | added | MLT | @IanMorris Could you please explain if it is possible to define a basis for the Grassmannian space $Gr(r,n)$? Also is there a innerproduct too? Thanks. | |
| Dec 4, 2013 at 14:52 | comment | added | A Blumenthal | @IanMorris Thank you. I actually suspect it isn't true in Banach space. I can show, though, that these two gaps aren't off by more than a constant factor in $\dim V = \dim W$ (one may use John's theorem to construct a 'good' basis and then fish around for the right inequality). | |
| Dec 4, 2013 at 14:17 | comment | added | Ian Morris | @ABlumenthal: in a Hilbert space this follows from Theorem I.6.34, although some translation is required to get the above statement. (My original post could be rather clearer in this respect.) One must use the fact that $d(V,W)$ equals the norm distance between the orthogonal projections onto $V$ and $W$ respectively (for which result one must refer to Akhiezer and Glazman) and then unpack a couple of definitions, to arrive at this statement. See Lemma 3.2 in bit.ly/18CoDdZ . In a Banach space I have no idea if the result is true or not. | |
| Dec 2, 2013 at 22:49 | comment | added | A Blumenthal | @IanMorris Where in Kato's book does he state that for subspaces of the same, finite dimension, the two gaps achieve their maxima simultaneously, and if so, is it stated for subspaces in a Banach space? | |
| Dec 7, 2010 at 22:12 | vote | accept | CommunityBot | ||
| Dec 6, 2010 at 19:18 | history | edited | Ian Morris | CC BY-SA 2.5 |
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| Dec 6, 2010 at 19:13 | comment | added | Ian Morris | Oh dear, I've really messed this up - why do I get out of bed some days? I'll try and fix it again, this time looking up the definition instead of trying to remember it! | |
| Dec 6, 2010 at 19:03 | comment | added | BS. | Do I miss something if I say that when $V\cap W$ is nonzero, your $d(V,W)$ vanishes ? | |
| Dec 6, 2010 at 19:00 | comment | added | AndrewLMarshall | I thought of the case inf was meant, but then k-planes with non-trivial intersections are distance 0 from each other. Shouldn't it be, maybe, inf over w, sup over v, ||v||=1, similarly to how the Hausdorff metric is defined? | |
| Dec 6, 2010 at 18:38 | comment | added | Ian Morris | Aha! I knew I wouldn't be able to write a post that long without making an elementary mistake. They should be infima, not suprema. (I've now fixed it.) | |
| Dec 6, 2010 at 18:32 | history | edited | Ian Morris | CC BY-SA 2.5 |
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| Dec 6, 2010 at 18:17 | comment | added | AndrewLMarshall | How are any of the sets above, which are being sup'ed over, bounded? In each expression one vector can have arbitrary magnitude, no? I don't understand. | |
| Dec 6, 2010 at 16:40 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Dec 6, 2010 at 14:13 | history | edited | Ian Morris | CC BY-SA 2.5 |
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| Dec 6, 2010 at 13:52 | history | edited | Ian Morris | CC BY-SA 2.5 |
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| Dec 6, 2010 at 13:38 | history | edited | Ian Morris | CC BY-SA 2.5 |
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| Dec 6, 2010 at 13:33 | history | answered | Ian Morris | CC BY-SA 2.5 |