= Gaussian independence of linear and quadratic statistics
{c}
{title2=$Y\sim N(\mu,\sigma^2I),\quad A=LL^T,\quad BA=0\Longrightarrow BY\perp Y^TAY$}
When $L$ has full column rank, its <Gram matrix> $L^TL$ is <positive-definite> and $BLL^TL(L^TL)^{-1}=BL=0$. The stacked normal vector $(BY,L^TY)$ therefore has zero cross <covariance>, so its blocks are independent. Since $Y^TAY=\|L^TY\|^2$, the linear and quadratic statistics are independent. This is the projection argument behind independence of <regression coefficient> and <variance> estimators in a <normal linear model>.
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