Specifically, assumeVis the covariance matrix,Eits eigenvector matrix, andDthe diagonal square root matrix of eigenvalues (on the diagonal), the update equation is

x_{i+1} =x_i +EDZ'

whereZis a matrix of independent rows(columns) of random standard normal variates. The other approach I have seen is to use

x_{i+1} =x_i + Sqrt(V)Z'

Is there a definition or theorem that would explain use of such square root matrices for quasi-Newton methods? What would the Z' matrix do to the step direction?