Fitted-residual orthogonality ($H(I-H)=0$)

For a full-rank ordinary least squares design matrix, the hat matrix $H$ and residual projection $G=I-H$ satisfy $HG=GH=0$. Hence fitted values and regression residuals are perpendicular as observed vectors. Under isotropic errors their cross-covariance matrix is zero, and under a normal linear model the two vectors are independent.