Page 429 - Matrix Analysis & Applied Linear Algebra

5.12 Singular Value Decomposition 425
Exercises for section 5.12

5.12.1. Following the derivation in the text, find an SVD for

−4 −6
C = .
3 −8
5.12.2. If σ 1 ≥ σ 2 ≥· · · ≥ σ r are the nonzero singular values of A, then it can
2 2 2 1/2
be shown that the function ν k (A)= σ + σ + ··· + σ k defines a
2
1
m×n m×n
unitarily invariant norm (recall Exercise 5.6.9) for (or C )
for each k =1, 2,...,r. Explain why the 2-norm and the Frobenius
2
norm (p. 279) are the extreme cases in the sense that A = σ 2 1 and
2
2
2
2
2
A = σ + σ + ··· + σ .
F 1 2 r
5.12.3. Each of the four common matrix norms can be bounded above and below
by a constant multiple of each of the other matrix norms. To be precise,
A ≤ α A , where α is the (i, j)-entry in the following matrix.
i j
1 2 ∞ F
 √ √ 
1 ∗ n n n
√ √
2  n ∗ n 1 
 √ √  .
∞  n n ∗ n 
√ √ √
F n n n ∗
For analyzing limiting behavior, it therefore makes no difference which
of these norms is used, so they are said to be equivalent matrix norms. (A
similar statement for vector norms was given in Exercise 5.1.8.) Explain
why the (2,F) and the (F, 2) entries are correct.
5.12.4. Prove that if σ 1 ≥ σ 2 ≥· · · ≥ σ r are the nonzero singular values of a
rank r matrix A, and if E <σ r , then rank (A + E) ≥ rank (A).
2
Note: This clarifies the meaning of the term “sufficiently small” in the
assertion on p. 216 that small perturbations can’t reduce rank.

5.12.5. Image of the Unit Sphere. Extend the result on p. 414 concerning
the image of the unit sphere to include singular and rectangular matrices
by showing that if σ 1 ≥ σ 2 ≥ ··· ≥ σ r > 0 are the nonzero singular
m
values of A m×n , then the image A(S 2 ) ⊂ of the unit 2-sphere
n th
S 2 ⊂ is an ellipsoid (possibly degenerate) in which the k semiaxis
is σ k U ∗k = AV ∗k , where U ∗k and V ∗k are respective left-hand and
right-hand singular vectors for A.