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Oct 1, 2018 at 1:40 history edited Mark L. Stone CC BY-SA 4.0
Corrected typo
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
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
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
May 31, 2018 at 20:11 history edited Mark L. Stone CC BY-SA 4.0
Added additional counterexample
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