Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors:
$\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + $\mathbf{x}_{1}^{\perp}$, where: $\mathbf{x}_{1}^{\text{T}} \mathbf{x}_{1}^{\perp}$ = $0$.
Can the quadratic form $\mathbf{x}_{0}^{\text{T}} \mathbf{A} \mathbf{x}_{0}$ be expressed in such a way that the quotient:
$s$ $=$ $\cfrac{\mathbf{x}_{0}^{\text{T}} \mathbf{A} \mathbf{x}_{0}}{\mathbf{x}_{1}^{\text{T}} \mathbf{A} \mathbf{x}_{1}}$
has a known (documented) distribution? I think a unitary matrix transformation per the helpful comment below, in addition to Cochran's Theorem, may be of utility here.