Prove Theorem A.3: Let Y ∼ Nn(????, ????), ???? > 0. Vector BY, with B a real q × n matrix, is independent of Y′

AY if B????A = ????.

a) First prove it in the case for which ???? = I by setting A = UDU′

, where U is orthogonal, D = diag([????1,…, ????k , 0,…, 0]) and k = rank(A). Let Z = U′

Y and B∗ = BU, and show that B∗D = ????.

b) Prove the general case by setting Z = ????−1∕2Y, B∗ = B????1∕2, and using the previous special case.