Sum of mutually orthogonal projectors ($P_i{P}_j={\delta }_{ij}{P}_i$)

If idempotent matrices $P_1,\dots ,P_k$ satisfy $P_i{P}_j=0$ whenever $i\ne j$, then $({\sum }_i{P}_i{)}^2={\sum }_{i,j}{P}_i{P}_j={\sum }_i{P}_i$. Thus the sum is idempotent. For a real orthonormal set $q_i$, the matrices $P_i=q_i{q}_i^T$ meet this condition. Their sum is the orthogonal projector onto their span; the symmetric matrix property follows by transposing the sum. Arbitrary sums of idempotent matrices need not be idempotent.