Can someone kindly explain idea behind the sum of projections? Here is my question, which is part of the beginning of the chi-squared decomposition theorem proof :
Suppose that $P_1, ....P_k $ are symmetric projection matices with sum the identity:

$I = P_1 + P_2 + ... + P_k $

Then squaring,

$ I = I^2 = \sum_i P_i^2 + \sum_{i&lt;j} P_i P_j $

My question is why after squaring, it yields the term $\sum_{i &lt;j} P_i P_j$ , how is it derived ???
Actually, under your conditions, $$I=I^2=\sum_{_i}P_i + 2\sum_{i&lt;j}P_i P_j$$
because $P_i^2 = P_i$ (all projection matrices are idempotent) and because $P_i$ , $P_j$ are symmetric with the same dimensions, so $P_i P_j + P_j P_i = 2 P_i P_j$.

This is simple algebra $(P_1+...+P_n)^2=(P_1+...+P_n)(P_1+...+P_n)$.
Actually, under your conditions, $$I=I^2=\sum_{_i}P_i + 2\sum_{i&lt;j}P_i P_j$$
because $P_i^2 = P_i$ (all projection matrices are idempotent) and because $P_i$ , $P_j$ are symmetric with the same dimensions, so $P_i P_j + P_j P_i = 2 P_i P_j$.

This is simple algebra $(P_1+...+P_n)^2=(P_1+...+P_n)(P_1+...+P_n)$.