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

Question

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:

• square root, then take the inverse of the square root: Inverse Square Root Of Matrix
• reciprocal of the square root of each term in the diagonal: Raising a square matrix to a negative half power
• plus tried (^-1/2) directly on multiple online matrix calculators

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

Answer 1

You can obtain the square root of a matrix M using the Cholesky Decomposition, M = LL'. Then compute the inverse of L.

Answer 2

@Avv, if $S=U L U^{T}$, then $S^{-1}=(U L U^{T})^{-1} = U L^{-1} U^{T} = U L^{-1/2} L^{-1/2} U^{T} = U L^{-1/2} U^T U L^{-1/2} U^{T} = (U L^{-1/2} U^T) (U L^{-1/2} U^{T})$,

thus $S^{-1/2} = U L^{-1/2} U^T$

Note that $U$ is orthonormal, and $L$ is diagonal