Page 287 - Matrix Analysis & Applied Linear Algebra

5.2 Matrix Norms 283

Properties of the 2-Norm
In addition to the properties shared by all induced norms, the 2-norm
enjoys the following special properties.
∗
• A = max max |y Ax|. (5.2.9)
2
x =1 y =1
2 2
∗
• A = A 2 . (5.2.10)
2
2
∗
• A A 2 = A . (5.2.11)
2

A
• 0
. (5.2.12)

= max A , B

0 B 2 2
2
∗
∗
∗
• U AV 2 = A 2 when UU = I and V V = I. (5.2.13)
You are asked to verify the validity of these properties in Exercise 5.2.6
on p. 285. Furthermore, some additional properties of the matrix 2-norm are
developed in Exercise 5.6.9 and on pp. 414 and 417.
Now that we understand how the euclidean vector norm induces the matrix
2-norm, let’s investigate the nature of the matrix norms that are induced by the
vector 1-norm and the vector ∞-norm.
Matrix 1-Norm and Matrix ∞-Norm
The matrix norms induced by the vector 1-norm and ∞-norm are as
follows.

• A = max Ax = max |a ij |
1 1
x =1 j (5.2.14)
1
i
= the largest absolute column sum.

• A = max Ax = max |a ij |
∞ =1 ∞ i
x (5.2.15)
∞
j
= the largest absolute row sum.
Proof of (5.2.14). For all x with x =1, the scalar triangle inequality yields
1

Ax = A i∗ x = a ij x j ≤ |a ij ||x j | = |x j | |a ij |

1
i i j i j j i

≤ |x j | max |a ij | = max |a ij | .
j j
j i i