When are $X^tC_1 X$ and $X^tC_2X$ independent?

Let $X$ be a $n\times p$ data matrix from $N_p(0,\Sigma)$. Let $C_1$ and $C_2$ be two symmmetric idempotent matrices. When are $X^tC_1 X$ and $X^tC_2X$ independent?

• When $C_1*C_2$=0 if I remember correctly – Deep North May 28 '17 at 12:34
• @DeepNorth Can you please explain how to derive that. ? Or refer sometext with that derivation? – Qwerty May 28 '17 at 12:42
• See this Craig's theorem www2.econ.iastate.edu/classes/econ671/hallam/documents/… – Deep North May 28 '17 at 12:45
• Hint: since the components of $X$ form an $np$-variate Normal distribution, those expressions are independent provided $C_1X_i$ and $C_2 X_j$ are uncorrelated for any rows $i,j$ of the matrix. Since all components have zero means, showing lack of correlation amounts to demonstrating $E(C_1X_i (C_2 X_j)^\prime)$ is the $n\times n$ zero matrix. That's a simple calculation. Note that it produces a result like the one quoted by @DeepNorth only when $\Sigma$ is diagonal. – whuber May 28 '17 at 14:34