$Y = \Sigma^{-1/2}(X-\mu )$ with $\Sigma$ being the covariance matrix and $\mu$ the sample mean.
What is the intuition behind this calculation? What kind of result does it yield?
This calculation is used in various estimates for multivariate Skewness.
As there is no information regarding the distribution of the random vector $X$, I assume that $$X\sim N_{p}(\mu,\Sigma),\qquad \Sigma >0.$$
The matrix $\Sigma^{-1/2}$ is the symmetric positive definite square root of $\Sigma$.
The transformation $Y=\Sigma^{-1/2}(X-\mu)$, results in a random vector whose components are independent and each is an univariate $N_{1}(0,1)$ random variable. In other words, the random vector $Y$ contains independent and identically distributed $N_{1}(0,1)$ random variables.
