Consider the ellipsoid (E=left{boldsymbol{x} in mathbb{R}^{d}: x boldsymbol{Sigma}^{-1} boldsymbol{x}=1 ight}) in (4.42). Let (mathbf{U D}^{2} mathbf{U}^{top}) be

Consider the ellipsoid \(E=\left\{\boldsymbol{x} \in \mathbb{R}^{d}: x \boldsymbol{\Sigma}^{-1} \boldsymbol{x}=1\right\}\) in (4.42). Let \(\mathbf{U D}^{2} \mathbf{U}^{\top}\) be an SVD of \(\boldsymbol{\Sigma}\). Show that the linear transformation \(\boldsymbol{x} \mapsto \mathbf{U}^{\top} \mathbf{D}^{-1} \boldsymbol{x}\) maps the points on \(E\) onto the unit sphere \(\{z\) \(\left.\in \mathbb{R}^{d}:\|z\|=1\right\}\).