SVD decomposition of orthonormal matrix Issue

Question

If i have a matrix $E \in \Re^{nxm}$ with orthonormal columns (e.g $E^TE = I_m$) then how to demonstrate (through Singular Value decomposition on $E$) that :

$I_n-EE^T = U(I_n -\Sigma\Sigma^T)U^T$ ?

PS : through SVD i establish the relation : $E^TE = VV^T$

Thanks for the help

Answer 1

If you have the SVD: $E=U\Sigma V$ then

\begin{align} I_n-EE^T &=I_n-U\Sigma V(U\Sigma V)^T\\ &=I_n-U\Sigma (VV^T)\Sigma^TU^T\\ &=UI_nU^T-U\Sigma (VV^T)\Sigma^TU^T\\ &= U(I_n -\Sigma\Sigma^T)U^T \end{align}