# Sphering data with SVD components of covariance matrix

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?

• Have you tried working out what the covariance of $X^{\star}$ will be? – cardinal Feb 8 '12 at 3:42

## 1 Answer

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.

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

$$= I$$