= Rank-sum form of Cochran's theorem
{title2=$\sum_i A_i=I,\quad\sum_i\operatorname{rank}A_i=n$}
Let $A_i$ be real <symmetric matrices> with sum $I_n$. Their <matrix ranks> sum to at least $n$, because their ranges span $\mathbb R^n$. If the ranks sum to $n$, those ranges form a <direct sum>. For $y$ in the range of $A_i$, the identity $y=\sum_jA_jy$ and uniqueness of its range decomposition imply $A_jy=\delta_{ij}y$. Thus $A_i^2=A_i$ and $A_iA_j=0$ for $i\ne j$: the matrices are mutually orthogonal <orthogonal projections>. In a combined <orthonormal basis>, the forms $Z^TA_iZ$ for $Z\sim N_n(0,I_n)$ use disjoint sets of independent standard normal coordinates. They are independent <chi-squared distributions> with degrees of freedom equal to the ranks. Conversely, if all these forms have those chi-squared laws, their expectations and their sum give $\sum_i\operatorname{rank}A_i=n$.
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