Timeline for On a condition for a matrix sum to be zero
Current License: CC BY-SA 4.0
16 events
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Oct 1, 2018 at 1:40 | history | edited | Mark L. Stone | CC BY-SA 4.0 |
Corrected typo
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Jun 2, 2018 at 4:15 | comment | added | Ludwig | @MarkL.Stone: Thanks for your comment! I would be really interested in providing an answer to my question in this particular case. So if you come up with an analytical counterexample, please let me know. | |
Jun 2, 2018 at 0:28 | comment | added | Mark L. Stone | @Ludwig The best I've achieved pertaining to my comment immediately above is having a maximum absolute element deviation from what is supposed to be the zero matrix of 1e-12; all other requirements are met with huge amount to spare. I strongly suspect that it is a valid counterexample.but don't have absolute certainty. As for any of m,n,N > 2, perhaps more is possible due to more degrees of freedom. | |
Jun 1, 2018 at 20:33 | comment | added | Mark L. Stone | @Ludwig I think counterexample can be done with m=n=N=2, A and B full rank, $Y_i$'s not identity, and A (say) upper triangular, but not also B upper triangular that I know of. Matrices are all messy, non-integer, and haven't tried to clean them up. | |
Jun 1, 2018 at 18:46 | vote | accept | Ludwig | ||
Jun 1, 2018 at 18:46 | comment | added | Ludwig | @MarkL.Stone: Excellent! Thank you! Out of curiosity, I'm wondering whether a counterexample can be found if we further assume that $A$ and $B$ are (lower) triangular matrices. | |
Jun 1, 2018 at 17:25 | history | edited | Mark L. Stone | CC BY-SA 4.0 |
added 675 characters in body
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Jun 1, 2018 at 16:29 | comment | added | Mark L. Stone | @Ludwig I added an exact counterexample with all matrix elmeents being integers. | |
Jun 1, 2018 at 16:28 | history | edited | Mark L. Stone | CC BY-SA 4.0 |
added 775 characters in body
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Jun 1, 2018 at 15:55 | comment | added | Ludwig | @MarkL.Stone: Thanks for sharing your numerical findings! However, I'd like to find an analytical counterexample in order to be 100% sure that the answer to my question is in the negative. | |
May 31, 2018 at 20:30 | comment | added | Mark L. Stone | @Federico Polon I have added a nicer counterexample, but I still stand by the first one. | |
May 31, 2018 at 20:20 | history | edited | Mark L. Stone | CC BY-SA 4.0 |
Added singular values of Y1 and Y2 in original; counterexample
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May 31, 2018 at 20:11 | history | edited | Mark L. Stone | CC BY-SA 4.0 |
Added additional counterexample
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May 31, 2018 at 14:36 | comment | added | Mark L. Stone | @Federico Poloni You may have misinterpreted my post. I was showing that a matrix which should be all zeros is, within numerical roundoff. I showed matrices which shouldn't be zero matrix are not zero matrix.I have checked this. Taking my "internal" A, and adding 5r-17 to all its elements, drives largest element deviation from zero of Y1*Y1'*Delta1*Y1*Y1'+Y2*Y2'*Delta2*Y2*Y2' from 2e-16 to 2e-13 | |
May 31, 2018 at 14:26 | comment | added | Federico Poloni | It's so small that it is not clear at all that it is not just a numerical error... | |
May 31, 2018 at 14:17 | history | answered | Mark L. Stone | CC BY-SA 4.0 |