Suppose we have a $k$-dimensional random variable $x$, and for any given idempotent matrix $A$, $x'Ax$ and $x'(I-A)x$ are independent to each other. Is $x$ normally distributed?
Far as I know, for $2$-dimensional $x$ and idempotent and symmetric $A$, once $Ax$ is always independent to $(I-A)x$, $x$ must have the $2$-dimensional normal distribution. And I think it's also true for higher dimensions.
I know the fact that an idempotent matrix can always be diagonalized by an orthogonal matrix to a diagonalized matrix with eigenvalues $0,1$. But I know little about the quadratic form.
Thanks in advance.