Page 414 - Matrix Analysis & Applied Linear Algebra

410 Chapter 5 Norms, Inner Products, and Orthogonality

5.11.6. Explain why the rank plus nullity theorem on p. 199 is a corollary of the
orthogonal decomposition theorem.

5.11.7. Suppose A = URV T is a URV factorization of an m × n matrix of

rank r, and suppose U is partitioned as U = U 1 | U 2 , where U 1
is m × r. Prove that P = U 1 U T 1 is the projector onto R (A) along

N A T . In this case, P is said to be an orthogonal projector because its
range is orthogonal to its nullspace. What is the orthogonal projector

onto N A T along R (A)? (Orthogonal projectors are discussed in
more detail on p. 429.)

5.11.8. Use the Householder reduction method as described in Example 5.11.2
to compute a URV factorization as well as orthonormal bases for the

−4 −2 −4 −2
four fundamental subspaces of A = 2 −2 2 1 .
−4 1 −4 −2
5.11.9. Compute a URV factorization for the matrix given in Exercise 5.11.8 by
using elementary row operations together with Gram–Schmidt orthogo-
nalization. Are the results the same as those of Exercise 5.11.8?

5.11.10. For the matrix A of Exercise 5.11.8, find vectors x ∈ R (A) and
T
y ∈ N A T such that v = x + y, where v =( 333 ) . Is there
more than one choice for x and y?

5.11.11. Construct a square matrix such that R (A) ∩ N (A)= 0, but R (A)is
not orthogonal to N (A).

5.11.12. For A n×n singular, explain why R (A) ⊥ N (A) implies index(A)=1,
but not conversely.

5.11.13. Prove that real-symmetric matrix ⇒ hermitian ⇒ normal ⇒ (com-
plex) RPN. Construct examples to show that none of the implications
is reversible.

5.11.14. Let A be a normal matrix.
(a) Prove that R (A − λI) ⊥ N (A − λI) for every scalar λ.
(b) Let λ and µ be scalars such that A − λI and A − µI are
singular matrices—such scalars are called eigenvalues of A.
Prove that if λ = µ, then N (A − λI) ⊥ N (A − µI).