Timeline for (Expected) Size of smallest singular value of a Vandermonde matrix associated to roots of polynomial
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Sep 1, 2017 at 17:17 | comment | added | Synia | Other suggestion : if $ s_1 $ is the smallest singular value, you have $ s_1 = \min_{ | x | = 1 } || V x ||^2 $ since this is the min of the quadratic form $ q_V(x) := x^* A x $ with $ A = V^* V $. You can play with this definition by particularising to interesting $x$, or even use the fact that $ s_1 \leq \int_B q_V(x) dx $ where $B$ is the unit ball and $ dx $ the Haar measure (not sure it gives something, though). The second moment method (Markov-Bienaymé-Tchebychev) tells you to look for the expectation of the square of $ \min_{|x| = 1} x^*(A - I) x $ since you want to compare to $1=|x|$. | |
| Sep 1, 2017 at 15:45 | comment | added | user70925 | The problem is that these kind of deterministic bounds only yields exponentially small bounds... Which are too far from 1 to explain this strange behaviour on expectations... | |
| Aug 31, 2017 at 1:01 | comment | added | Synia | You can try any of the deterministic bounds given in the following article : library.utia.cas.cz/separaty/2009/AS/… After what, you can take the expectation. You may have to use Jensen to bound from below this last expectation. | |
| Aug 30, 2017 at 13:32 | history | asked | user70925 | CC BY-SA 3.0 |