If the matrix \( \boldsymbol{X} \) is an orthogonal (or unitary in case of complex values) matrix, we have
$$ \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{X}\boldsymbol{X}^T = \boldsymbol{I}. $$In this case the matrix \( \boldsymbol{A} \) becomes
$$ \boldsymbol{A}=\boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T)=\boldsymbol{I}, $$and we have the obvious case
$$ \boldsymbol{\epsilon}=\boldsymbol{y}-\tilde{\boldsymbol{y}}=0. $$This serves also as a useful test of our codes.