8.36. (Sec. 8.10.2) Monotonicity of the power function of the Bartlett-Nanda-Pillai trace test. Show that tr'(uu' +B)(uu' +B+ W)-l 5,K is convex in u for fixed positive semidefinite B and positive definite B + W if 05, K 5, 1. [Hint: Verify

(UU'+B+W)-I

=(B+W)'I- 1 _I (B+W)-IUU'(B+W)-I.

l+u'(B+W) u The resulting quadratic form in u involves the matrix (tr A)l- A for A =

(B + W)- tB(B + W)- 1; show that this matrix is positive semidefinite by diagonalizing A.]