STAT 714, FALL 2008 HOMEWORK 2
1. For any matrix A, show that R(A0A) = R(A).
2. Define the matrix
A=
1100
1100
0010
0011
.
(a) Find C(A) and a basis for this space.
(b) Find N(A0) and a basis for this space.
(c) Find 2 different generalized inverses of A, say A−
1and A−
2. Check your work by
showing that AA−
1A=Aand AA−
2A=A.
(d) Compute MA, the perpendicular projection matrix onto C(A).
(e) Compute I−MA, the perpendicular projection matrix onto N(A0).
3. Let An×p,bp×1,cn×1, and suppose that the equations Ab =care consistent. Let
xn×1,up×1, and Xp×n. Let A−
1and A−
2be two generalized inverses of A. Let Idenote
the n×nidentity matrix.
(a) Let b∗be a solution to Ab =c. Show that b∗+uc0{(A−
1)0A0−I}xis also a solution.
(b) Show that A−
1+X(AA−
2−I) is a generalized inverse of A.
4. (a) Suppose that Ais an n×nsymmetric matrix. Prove that Ais idempotent if and
only if r(A) + r(I−A) = n.
(b) Suppose that A,B, and A+Bare all idempotent. Prove that AB =0and BA =0.
5. Let Pbe an n×northogonal matrix and let Abe an n×nsymmetric and idempotent
matrix. Define D=P0AP.
(a) Show that Dis a perpendicular projection matrix.
(b) Show that if Ais nonnegative definite, then so is D.
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