Let X be an n × p matrix with rank p . Suppose the first column of X is 1, a column of 1s. Then, show that (a) P1 = 1. (b) (J/n), P − J/n, and (I − P ) are idempotent and pairwise orthogonal, where P...

Let X be an n × p matrix with rank p . Suppose the first column of X is 1, a column of 1s. Then, show that

(a) P1 = 1.

(b) (J/n), P − J/n, and (I − P ) are idempotent and pairwise orthogonal, where P = X(X’X)−1X and J/n is given in equation 4.22.