1. The problem statement, all variables and given/known data Trying to show that if a matrix A=A_1+...+A_n is idempotent and that if the sum of the ranks of A_i is equal to the rank of A then each A_i is idempotent. The A_i's are symmetric and real, of course. This isn't really a homework question per se, I am writing notes for a reading class and I am trying to show that a particular distribution is chi^2 and I decided to add an appendix with cochrans theorem. 2. Relevant equations http://www.stat.columbia.edu/~fwood/Teaching/w4315/Fall2009/lecture_cochran.pdf I found this but the extension from beyond n=2 is not that obvious. 3. The attempt at a solution It's obvious that A has an orthogonal decomposition of the form A=V^T diag(I_r,0)V and each of the A_i's have a similar decomposition by the spectral theorem. I realize that the rank will be unchanged by such an orthogonal decomposition. Each A_i will decompose into V_i^T diag(eigen1,eigen2,...,eigen(rank A_i),0,...) V_i. Any nudge in the right direction would be appreciated.