Let y be a vector with a a ay ay and y 1

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 p-norms 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...
View Full Document

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.

Ask a homework question - tutors are online