Random Matrix Theory - Semicircle Law - Non positive definite matrices

almostcutmyhair · July 24th, 2014, 7:09 pm

Is there a way to utilize Wigner's semicircle law to approximate non-positive-definite (NPD) matrices with closest positive-definite (PD) ones? In other words, is there a sensible way to set the nonpositive eigenvalues of a NPD matrix to positive ones using random matrix theory?

paulptli · July 25th, 2014, 1:17 pm

What makes you think you can use it?

almostcutmyhair · July 25th, 2014, 3:12 pm

Perhaps by determining a cutoff point for the eigenvalues by somehow finding where the noise starts?

Alan · July 25th, 2014, 5:50 pm

Perhaps the problem of determining the closest positive definite matrix to a given matrix is already solved?See, for example the discussion here

paulptli · July 27th, 2014, 2:37 pm

Do you realize, that matrices there are NOT necessarily positive definite, and the fact that limiting distribution is sometimes drawn on [0,1] is just normalization?

almostcutmyhair · July 29th, 2014, 11:18 pm

QuoteOriginally posted by: AlanPerhaps the problem of determining the closest positive definite matrix to a given matrix is already solved?See, for example the discussion hereThat's only one approach, and a very time consuming approach.

almostcutmyhair · July 29th, 2014, 11:19 pm

QuoteOriginally posted by: paulptliDo you realize, that matrices there are NOT necessarily positive definite, and the fact that limiting distribution is sometimes drawn on [0,1] is just normalization?Can you rephrase this?

quartz · July 30th, 2014, 8:53 am

QuoteOriginally posted by: almostcutmyhairPerhaps by determining a cutoff point for the eigenvalues by somehow finding where the noise starts?One usually wants to avoid negative eigenvalues regardless of where the RMT cutoff lies. Moreover your actual distribution is probably not normal, so that a cutoff from classic RMT is not even correct (of course there are extensions).