Unformatted text preview: Ay as a linear combination of columns
of A, see Section 1.5, and applying the triangle inequality for vector norms
gives
A 1 = Ay 1 = y1 Ae1 + · · · + yn Aen 1 ≤ y1  Ae1
≤ (y1  + · · · + yn ) max Aej 1 . 1 + · · · + yn  Aen 1 ≤j ≤n From y1  + · · · + yn  = y 1 = 1 follows A 1 ≤ max1≤j ≤n Aej The matrix inﬁnity norm is equal to the maximal absolute row sum.
Fact 2.17 (Inﬁnity Norm). Let A ∈ Cm×n . Then
n A ∞ ∗ = max A ei
1 ≤i ≤m 1 = max 1 ≤i ≤m j =1 aij . 1. 1 Downloaded 01/17/14 to 143.215.200.123. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 2.6. Matrix Norms 35 ∗
Proof. Denote the rows of A by ri∗ = ei A, and let rk have the largest one norm,
rk 1 = max1≤i ≤m ri 1 . • Let y be a vector with A
A ∞ = Ay ∞ ∞ = Ay and y ∞ = 1. Then ∞ = max ri∗ y  ≤ max ri
1≤i ≤m y 1 1 ≤i ≤m ∞ = rk 1, where the inequality follows from Fact 2.8. Hence A ∞ ≤ max1≤i ≤ ri 1 .
∗
• For any vector y with y ∞ = 1 we have A ∞ ≥ Ay ∞ ≥ rk y . Now
∗
we show how to choose the elements of y such that rk y = rk 1 . Let
∗
∗
rk = ρ1 . . . ρn be the elements of rk . Choose the elements of y such
that ρj yj = ρj . That is, if ρj = 0, then yj = 0, and otherwise yj = ρj /ρj .
∗
Then y ∞ = 1 and rk y  = n=1 ρj yj = n=1 ρj  = rk 1 . Hence
j
j
A ∞ ∗
≥ rk y  = rk 1 = max ri
1 ≤i ≤m 1. The pnorms satisfy the following submultiplicative inequality.
Fact 2.18 (Norm of a Product). If A ∈ Cm×n and B ∈ Cn×p , then
AB
Proof. Let x ∈ Cp such that AB
2.15 twice gives
AB p = ABx p ≤A p p
p ≤A B p p. = ABx p
and x ≤A p B Bx p p p x p = 1. Applying Remark
=A p B p. Since the computation of the two norm is more involved, we postpone it
until later. However, even without knowing how to compute it, we can still derive
several useful properties of the two norm. If x is a column vector, then x 2 = x ∗ x .
2
We show below that an analogous property holds f...
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This note was uploaded on 01/30/2014 for the course MATH 1502 taught by Professor Mcclain during the Spring '07 term at Georgia Tech.
 Spring '07
 McClain
 Multiplication, Scalar

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