Sphering data with SVD components of covariance matrix

Question

The Elements of Statistical Learning says on page 113:

Sphere the data with respect to the common covariance estimates $\hat{\Sigma}$: $X^* \leftarrow D^{-1/2}U^TX$ where $\hat{\Sigma} = UDU^T$. The common covariance estimate of $X^*$ will now be the identity.

Can someone help me understand why $X^* \leftarrow D^{-1/2}U^TX$ spheres the data?

Answer 1

I think I figured out the answer after seeing cardinal's suggestion and reading the Wikipedia page on whitening.

$cov(X^*) = E[X^*X^{*T}]$

$= E[D^{-\frac{1}{2}}U^TXX^TUD^{-\frac{1}{2}T}]$

$D^{-\frac{1}{2}T} = D^{-\frac{1}{2}}$ because it's a diagonal matrix

$= D^{-\frac{1}{2}}U^TE[XX^T]UD^{-\frac{1}{2}}$

$= D^{-\frac{1}{2}}U^T\hat{\Sigma}UD^{-\frac{1}{2}}$

$= D^{-\frac{1}{2}}U^TUDU^TUD^{-\frac{1}{2}}$

The $U^TU = 1$ because U's have unit length.

$= D^{-\frac{1}{2}}DD^{-\frac{1}{2}}$

$= I$