Quasi-Newton method with square root matrix

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?