Restricted likelihood from orthogonal error contrasts (${\ell }_R$)

For $Y\sim N(X\beta ,V_{\theta })$ and a full-column-rank design matrix $X\in {\mathbb{R}}^{n\times p}$, take $A\in {\mathbb{R}}^{n\times (n-p)}$ with $A^{T}A=I$ and $A^{T}X=0$. Then $A^{T}Y\sim N(0,A^T{V}_{\theta }A)$, so restricted maximum likelihood maximizesChanging the orthonormal basis by an orthogonal matrix leaves this expression unchanged.