Page 288 - Matrix Analysis & Applied Linear Algebra

284 Chapter 5 Norms, Inner Products, and Orthogonality

Equality can be attained because if A ∗k is the column with largest absolute sum,

set x = e k , and note that e k =1 and Ae k = A ∗k = max j |a ij | .
1 1 1 i
Proof of (5.2.15). For all x with x =1,
∞

Ax = max a ij x j ≤ max |a ij ||x j |≤ max |a ij | .
∞
i i i
j j j
Equality can be attained because if A k∗ is the row with largest absolute sum,
and if x is the vector such that

1if a kj ≥ 0, |A i∗ x| = | j a ij x j |≤ j |a ij | for all i,
x j = then
−1if a kj < 0, |A k∗ x| = j |a kj | = max i j |a ij | ,

so x =1, and Ax = max i |A i∗ x| = max i |a ij | .
∞ ∞ j
Example 5.2.2
Problem: Determine the induced matrix norms A 1 and A ∞ for

1 3 −1
A = √ √ ,
3 0 8
and compare the results with A 2 (from Example 5.2.1) and A F .
Solution: Equation (5.2.14) says that A 1 is the largest absolute column sum
in A, and (5.2.15) says that A ∞ is the largest absolute row sum, so
√ √ √ √
A 1 =1/ 3+ 8/ 3 ≈ 2.21 and A ∞ =4/ 3 ≈ 2.31.
√
T
Since A 2 =2 (Example 5.2.1) and A F = trace (A A)= 6 ≈ 2.45, we
see that while A 1 , A 2 , A ∞ , and A F are not equal, they are all in
the same ballpark. This is true for all n × n matrices because it can be shown
that 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 ∗
(see Exercise 5.1.8 and Exercise 5.12.3 on p. 425). Since it’s often the case that
only the order of magnitude of A is needed and not the exact value (e.g.,
recall the rule of thumb in Example 3.8.2 on p. 129), and since A 2 is difficult
to compute in comparison with A 1 , A ∞ , and A F , you can see why any
of these three might be preferred over A 2 in spite of the fact that A 2 is
more “natural” by virtue of being induced by the euclidean vector norm.