Suppose X is an n×p matrix with rank p ≤ n.

(a) Show that X

X is p×p positive definite. Hint: if c is p×1, what is c

X

Xc?

(b) Show that XX is n×n non-negative definite.

For exercises 3–6, suppose R is an n×n orthogonal matrix and D is an n×n diagonal matrix, with Dii > 0 for all i. Let G = RDR

. Work the exercises directly, without appealing to theorem 1.