Show that the maximum-likelihood estimate of (beta) satisfies [ left[X^{prime} K^{prime}left(K V K^{prime}ight)^{-1} K Xight] hat{beta}=X^{prime} K^{prime}left(K

Question:

Show that the maximum-likelihood estimate of $\beta$ satisfies

\[
\left[X^{\prime} K^{\prime}\left(K V K^{\prime}ight)^{-1} K Xight] \hat{\beta}=X^{\prime} K^{\prime}\left(K V K^{\prime}ight)^{-1} Z=X^{\prime} W Q Y
\]

where $Q: \mathcal{H} ightarrow \mathcal{H}$ is the orthogonal projection with $\operatorname{kernel} \operatorname{ker}(K)$.

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