Principles of Mathematics in Operations Research

84 6 Computational Aspects

Definition 6.1.7 The norm of A is the number defined \\A\\ = xnaxx^o |d!f

Remark 6.1.8 ||.A|| bounds the "amplifying power" of the matrix.

\\Ax\\ < \\A\\ \\x\\, Vz;

and equality holds for at least one nonzero x. It measures the largest amount

by which any vector (eigen vector or not) is amplified by matrix multiplication.

Proposition 6.1.9 For a square nonsingular matrix, the solution x = A~^1 b

and the error Ax = A~^1 Ab satisfy

J!M<M||.u-i||JJM.

Proof. Since

b=Ax=> \\b\\ < \\A\\\\x\\ and

Ax = A-xAb =$> \\AX =|| < ||^_1|| ||A>||, we have

H&HPHIWI and II^Hl^-

1

!!!!^

Remark 6.1.10 When A is symmetric,

W = |A„I, ii^-

1

^ J-|=>c:

D

lAllIU-^1 !^

|Ai|

and the relative error satisfies

\AX\\ K \\Ab[

"1 K

0 1

,b =

K

1 , A» =

0"

-1

Example 6.1.11 Let us continue the previous example, where

A =

Since we have

«<||^||<«+1, and K < \\A~*\\ < K+ 1,

then the relative amplification is approximately K^2 « ||.A|| ||A_1|

Remark 6.1.12

ll2 \\Ax\\

(^2) xTATAx
\A\' = max

M

= max • : Rayleigh quotient!