= Sum of mutually orthogonal projectors
{title2=$P_iP_j=\delta_{ij}P_i$}
If <idempotent> <matrices> $P_1,\ldots,P_k$ satisfy $P_iP_j=0$ whenever $i\ne j$, then $(\sum_iP_i)^2=\sum_{i,j}P_iP_j=\sum_iP_i$. Thus the sum is <idempotent>. For a real <orthonormal set> $q_i$, the <matrices> $P_i=q_iq_i^T$ 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>.
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