 Solution

Assume first that $D=b^T{V}^{-1}b>0$. Since $X_M={\theta }_M^{T}Z$,and for $Y={\varphi }^{T}Z$,The least-squares affine-regression coefficients are thereforeThe minimum iswhereThis matrix is symmetric and positive semidefinite. If $b=0$, then $X_M=0$, one may take $\alpha =0$ and arbitrary $\beta$, and the minimum is ${\varphi }^{T}V\varphi$, corresponding to $Q=V$.