Linear Algebra, Theory And Applications, 2012a

14.2. THE CONDITION NUMBER 345
Thus
and so at this value of t,
t = bq/(p+q)
a p/(p+q)
( )
at =(ab) q/(p+q) b
, =(ab) p/(p+q) .
t
Thus the minimum of f is
1
(
(ab) q/(p+q)) p 1
(
+ (ab) p/(p+q)) q
pq/(p+q)
=(ab)
p
q
but recall 1/p +1/q = 1 and so pq/ (p + q) =1. Thus the minimum value of f is ab. Letting
t =1, this shows
ab ≤ ap
p + bq
q .
Note that equality occurs when the minimum value happens for t = 1 and this indicates
from (14.5) that a p = b q .
Now ||A|| p
may be considered as the operator norm of A taken with respect to ||·|| p
. In
thecasewhenp =2, this is just the spectral norm. There is an easy estimate for ||A|| p
in
terms of the entries of A.
Theorem 14.1.5 The following holds.
⎛ ⎛
⎜∑
||A|| p
≤ ⎝ ⎝ ∑ j
k
⎞ ⎞ q/p
1/q
|A jk | p ⎠ ⎟ ⎠
Proof: Let ||x|| p
≤ 1andletA =(a 1 , ··· , a n )wherethea k are the columns of A. Then
( ) ∑
Ax = x k a k
k
and so by Holder’s inequality,
||Ax|| p
≡
≤
∣∣ ∑ ∣∣∣∣ ∣∣∣∣ x
∣∣
k a k ≤ ∑ |x k |||a k || p
k p k
( ) 1/p ( ∑ ∑
|x k | p ||a k || q p
k
k
) 1/q
≤
⎛ ⎛
⎜
∑
⎝ ⎝ ∑ j
k
⎞ ⎞ q/p
1/q
|A jk | p ⎠ ⎟ ⎠
14.2 The Condition Number
Let A ∈L(X, X) be a linear transformation where X is a finite dimensional vector space
and consider the problem Ax = b where it is assumed there is a unique solution to this
problem. How does the solution change if A is changed a little bit and if b is changed a