A Bit of Numerical Analysis - Computer Graphics and Geometric Modeling - page 907

APPENDIX F

A Bit of Numerical Analysis

F.1 The Condition Number of a Matrix

F.1.1. Lemma. If A is an m ¥ n matrix, then there is a real number K > 0, so

that

x AK £ x

for all x ΠR m .

Proof. Assume A = (a ij ). Let c j = (a 1j ,a 2j ,...,a mj ) be the jth column vector and let

{ }

M = max cc , ,..., c .

12 n

If y = x A, then the Cauchy-Schwarz inequality implies that

m

 1

y = a x £ cx £ M x ,

i ij j i

i =

so that

2

2 2 2 2

xy A = £ y + y + ◊◊◊+ y £ nM x £ n Mx .

n

1 2

We can let K M = .

Definition. Let A be an m ¥ n matrix. The norm of A, denoted by ||A||, is defined by

x
x

A

A = sup .

n

0xR πŒ

Lemma F.1.1 clearly implies that the norm of an m ¥ n matrix is well defined
because a bounded set of real numbers always has a least upper bound. The next
theorem shows that the norm of a matrix behaves like a norm.