Page 289 - Matrix Analysis & Applied Linear Algebra

5.2 Matrix Norms 285
Exercises for section 5.2

5.2.1. Evaluate the Frobenius matrix norm for each matrix below.
   
010 4 −2 4
1 −2
A = , B =  001  , C =  −2 1 −2  .
−1 2
100 4 −2 4
5.2.2. Evaluate the induced 1-, 2-, and ∞-matrix norm for each of the three
matrices given in Exercise 5.2.1.

5.2.3. (a) Explain why I =1 for every induced matrix norm (5.2.4).
(b) What is I n×n ?
F
∗
5.2.4. Explain why A = A for Frobenius matrix norm (5.2.1).
F F
5.2.5. For matrices A and B and for vectors x, establish the following com-
p
patibility properties between a vector norm defined on every C and
the associated induced matrix norm.
(a) Show that Ax ≤ A x .
(b) Show that AB ≤ A B .
(c) Explain why A = max x ≤1 Ax .

5.2.6. Establish the following properties of the matrix 2-norm.
(a) A = max |y Ax|,
∗
2
x 2 =1
y 2 =1
(b) A = A 2 ,
∗
2
2
∗
(c) A A 2 = A ,
2

A
(d)
0
(take A, B to be real),

= max A , B
0 B 2 2
2
(e) U AV 2 = A 2 when UU = I and V V = I.
∗
∗
∗
5.2.7. Using the induced matrix norm (5.2.4), prove that if A is nonsingular,
then
1

1
A =

or, equivalently,
A −1
= .
min
A −1
min Ax
x
x =1 x =1
n×n −1
5.2.8. For A ∈C and a parameter z ∈C, the matrix R(z)=(zI − A)
is called the resolvent of A. Prove that if |z| > A for any induced
matrix norm, then
1
R(z) ≤ .
|z|− A