Deriving $F$ distributions

Here are some well known distributional results introducing the $\chi^2$ and the $F$ distributions in relation to GLM using the geometrical approach of projectors [12][3].
Let $y\sim N(\mu,\sigma^2V)$ where $\mu=X\beta$ with $X$ of full rank $n \times p$, $p<n$, i.e. $\mu \in [X]$ where $[X]$ means the space generated by the columns of $X$, then:

1. $\left\Vert P_Gy\right\Vert ^2/\sigma^2\sim \chi ^2(trace(V^-P_GV), \left\Vert P_G\mu\right\Vert ^2 /(2\sigma^2))$,
   where $P_G=G(^tGV^-G)^-{\;}^tGV^-$, is the orthogonal projector³¹ onto G, then $\left\Vert P_Gy\right\Vert ^2=^t(P_Gy)V^-P_Gy$ is the Sum of Squares due to G (SSG) in the metric space $(I\!\!R^n;V^-)$.³²
2. if $[G_1]\bot[G_2]$ then $\left\Vert P_{G_1}y\right\Vert ^2$ and $\left\Vert P_{G_2}y\right\Vert ^2$ are independent, then their ratio follows an $F$ distribution:
   
   $$\frac{trace(V^-P_{G_2}V)}{trace(V^-P_{G_1}V)} \frac{\left\Ve... ...}y\right\Vert ^2} \sim F(trace(V^-P_{G_1}V),trace(V^-P_{G_2}V)$$
   
   

3. testing $H_0:$ $\mu \in [X_0]\subset[X]$ with $rank(X_0)=k$, is equivalent to testing $P_{[X]\cap[X_0]^\bot}y=0$ (i.e. the projection of $y$ onto $[X]$ orthogonally to $[X_0]$ is null),
   then under $H_0$
   
   $$\frac{(trace(VV^-)-trace(P_X))}{(trace(P_X)-trace(P_{X_0})} ... ...^\bot}y\right\Vert ^2}{\left\Vert P_{[X]^\bot} y\right\Vert ^2}$$
   
   
   
   
   $$\sim F((trace(P_X)-trace(P_{X_0}),(trace(VV^-)-trace(P_X)))$$
   
   
   note that as $[X_0]\subset [X]$, $P_{[X]\cap[X_0]^\bot}=P_{[X]} -P_{[X_0]}$. Sometimes $[X]\cap[X_0]^\bot$ is noted $[X]/[X_0]$ and can be called a conditioned model, $[X_0]$ called the sub-model. The $F_0$ ratio can be written as $(SSX -SSX_0 /SSE)$ or $(SSE_0 -SSE)/SSE$ times the ratio of $df$. If $V=Id$ one finds the traditional $F_0 \sim F(p-k,n-p)$.