I'm observing some unexpected behavior in matlab after the following steps:
a) start with a non-definite symmetric matrix X (of n >= 10)
b) do an eigen decomposition of X and set all negative eigenvalues to 0
c) reconstruct X_hat and X_hat_inverse - which should be positive semi-definite.
d) check the eigenvalues of X_hat and X_hat_inverse
Both these matrices sometimes turn out to have negative (albeit very small) values !
Here's some code for that:
X = rand(1000,10); X = X'*X/1000; eig(X) % all positive X(X(:)<0.25) = 0 % no longer psd [uu_,dd_] = eig(X); dd_( dd_(:)<0 ) = 0; X_hat = uu_*dd_*uu_'; X_hat_inv = uu_*pinv(dd_)*uu_'; eig(X_hat) %negative e.v.s !! eig(X_hat_inv) % complex e.v.s !!!
are these acceptable numerical errors - or is something wrong ?
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