Timeline for The probability for a symmetric matrix to be positive definite
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 1, 2014 at 8:27 | vote | accept | Denis Serre | ||
| Mar 15, 2013 at 11:01 | history | edited | Mikael de la Salle | CC BY-SA 3.0 |
Added precise value of c and references; deleted 2 characters in body
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| Jan 10, 2013 at 20:48 | comment | added | Kevin P. Costello | One strange corollary of this: Imagine exposing your matrix minor by minor (so that after step $k$ the $k \times k$ upper left submatrix is exposed). By Sylvester's criterion, $M$ is positive definite iff the determinants of each exposed submatrix are positive. So what this is saying is that the probability the $n^{th}$ determinant is positive, conditioned on the previous determinants being positive, decays exponentially in $n$. I find this counterintuitive, especially given that individual entries have symmetric distribution. | |
| Jan 10, 2013 at 19:08 | comment | added | Mikael de la Salle | My guess (no proof) is that the non-tivial asymptotics will occur for the n-uple $(\log(p_{n,i})/n^2)_{i=0}^n$. This should also follow from the Ben Arous-Guionnet large deviation results. | |
| Jan 10, 2013 at 18:20 | comment | added | Denis Serre | You're right. If the limiting graph is identically zero (something I anticipated), then I am interesting in a non-trivial asymptotics, when we rescale this graph appropriately. | |
| Jan 10, 2013 at 17:46 | history | answered | Mikael de la Salle | CC BY-SA 3.0 |