Cochran's theorem: rank-sum equal to $n$ plus positive semi-definiteness gives independent chi-squares

Let $\underline X\sim N_n(\underline0,\sigma^2I_n)$. Let $Y=\sum_{i=1}^nX_i^2/\sigma^2$ and $Q_i=\underline X'A_i\underline X$ with rank$(A_i)=r_i, i=1,2,\dots,k$. Let $Y=\sum_{i=1}^kQ_i$. Then $Q_1,Q_2,\dots,Q_k$ are independently distributed if and only if: