# Matrices - Inverse of the principal square root of a covariance matrix (^-1/2)

Say you have a square (variance)covariance matrix S

How would one go about working S^-1/2 (inverse of the principle square)?

Bearing in mind, I'm trying to understand a paper which states:

I've tried multiple suggestions, such as:

However, none of them seem to hold true for the latter part of the what's stated on the paper.