Timeline for PSD matrix with non-negative entries
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2011 at 7:24 | comment | added | Denis Serre | I verified by hands that Hall's counter-example is valid. | |
| Jun 10, 2011 at 11:48 | comment | added | Noah Stein | I agree that Gray and Wilson's method now seems suspect, but I do not think they are claiming that such orthogonality is necessary for a counterexample, merely sufficient (though this too may be false). It is certainly not necessary, as there are counterexamples with all entries strictly positive. Since the set of completely positive matrices is closed, any matrix $A$ which is doubly nonnegative but not completely positive can be bounded away from the completely positive matrices. We can thus add a small enough $\epsilon$ to all entries of $A$ to get a strictly positive counterexample. | |
| Jun 10, 2011 at 4:40 | comment | added | Pawan Aurora | If the example due to Hall is correct (as it seems to be), then the scheme suggested by Gray & Wilson for creating counter examples for any $M$ seems to be flawed as well. Hall's example does not have any three vectors mutually orthogonal as required by Gray & Wilson. | |
| Jun 9, 2011 at 13:00 | history | edited | Noah Stein | CC BY-SA 3.0 |
added 148 characters in body
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| Jun 9, 2011 at 12:59 | comment | added | Noah Stein | @Denis Serre: Thanks for the counter-counterexample! I had been a bit suspicious of the method used to construct it, as it seemed to assume that the CP-rank of a matrix is equal to the rank, or something like that. Hopefully the original example due to Hall quoted by Suvrit is correct. | |
| Jun 9, 2011 at 9:33 | comment | added | Denis Serre | Now I have a doubt about the 'counter-example': This matrix writes as $$\begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\\\ 0 & 1 & 0 & 0 & 1 \\\\ 0 & 0 & 8/9 & 0 & 8 \\\\ 0 & 0 & 0 & 0 & 0 \\\\ 1 & 1 & 8 & 0 & 74 \end{pmatrix}+\begin{pmatrix} 1 & 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 &1 & 0 \\\\ 0 & 0 & 1/9 & 1 & 0 \\\\ 1 & 1 & 1 & 11 & 0 \\\\ 0 & 0 & 0 & 0 & 0 \end{pmatrix},$$ where both matrices are non-negative. Since the answer to the question is positive for $n=4$, and these matrices are block-diagonal, $A$ can be written in the requested form. | |
| Jun 9, 2011 at 8:15 | vote | accept | Pawan Aurora | ||
| Jun 9, 2011 at 6:58 | comment | added | Denis Serre | Good. The example is accurate in the sense that it is not positive definite. | |
| Jun 8, 2011 at 22:42 | comment | added | Noah Stein | @Suvrit: Interesting. I don't think I had seen that paper before, though I don't consider myself terribly familiar with the history of this area. Diananda's result is essentially the dual of the result requested here. It looks like Maxfield and Minc may have been first to put it in this form. | |
| Jun 8, 2011 at 22:39 | history | edited | Noah Stein | CC BY-SA 3.0 |
Corrected Horn to Hall
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| Jun 8, 2011 at 21:55 | comment | added | Suvrit | actually, some literature search reveals that the claim for $M \le 4$ is due to Maxfield and Minc in a 1962 paper; correct me if i'm wrong. | |
| Jun 8, 2011 at 19:56 | history | edited | Noah Stein | CC BY-SA 3.0 |
added 255 characters in body
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| Jun 8, 2011 at 19:47 | history | answered | Noah Stein | CC BY-SA 3.0 |