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