Specifically, assume V is the covariance matrix, E its eigenvector matrix, and D the diagonal square root matrix of eigenvalues (on the diagonal), the update equation is

x_{i+1} = x_i + EDZ'

where Z is 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?

Several reports stated that it's common to use the square root of the Hessian since its condition number (ratio of largest eigenvalue to smallest eigenvalue) is less severe.