Let Ml and M2 be perpendicular projection matrices, and let Mo be a perpendicular projection operator onto C(M1 ) n C(M2). Show that the following are equivalent:

(a) MIM2 = M2M1·

(b) MIM2 = Mo·

(c) {C(M1 ) n [C(M1 ) n C(M2)]J. } 1- {C(M2) n [C(Ml) n C(M2)]J. }.

Hints: (i) Show that MIM2 is a projection operator. (ii) Show that MIM2 is symmetric. (iii) Note that C(M1 ) n [C(Md n C(M2)]J. = C(MI - Mo).

Exercise B.15 ces. Show that Let Ml and M2 be perpendicular projection matri-

(a) the eigenvalues of MIM2 have length no greater than 1 in absolute value (they may be complex).

(b) tr(M1M2) ::; r(M1M2).

Hints: For part

(a) show that with x' M x == 11M x1l 2, 11M xii::; Ilxll for any perpendicular projection operator M. Use this to show that if M1M2x =

AX, then IIMIM2xll ~ IAIIIMIM2XII.