Vector Norms

CSE 541

Roger Crawfis

Vector Norms

Measure the magnitude of a vector Is the error in x small or large?

General class of p-norms:

1-norm: 2-norm: -norm:

1 1

n

iix x

1/ 22

2 1

n

iix x

max i ix x

ppn

ip

xx

1

1

Properties of Vector Norms

For any vector norm:

These properties define a vector norm

0 if 0

for any scalar

(triangle inequality)

x x

x x

x y x y

Matrix Norms

We will only use matrix norms “induced” by vector norms:

1-norm:

-norm:

0maxx

AxA

x

11

max (max absolute column sum)n

ijj

i

A A

1

max (max absolute row sum)n

iji

j

A A

Properties of Matrix Norms

These induced matrix norms satisfy:

0 if A 0

for any scalar

(triangle inequality)

for any vector

A

A A

A B A B

AB A B

Ax A x x

Condition Number

If A is square and nonsingular, then

If A is singular, then cond(A) = If A is nearly singular, then cond(A) is large. The condition number measures the ratio of

maximum stretch to maximum shrinkage:

1cond( )A A A

1

1

00max min

xx

Ax AxA A

x x

Properties of Condition Number

For any matrix A, cond(A) 1For the identity matrix, cond(I) = 1For any permutation matrix, cond(P) = 1For any scalar , cond( A) = cond(A)For any diagonal matrix D,

cond( ) max / minii iiD D D

Errors and Residuals

Residual for an approximate solution y to Ax = b is defined as r = b – Ay

If A is nonsingular, then ||x – y|| = 0 if and only if ||r || = 0. Does not imply that if ||r||<, then ||x-y|| is small.

Estimating Accuracy

Let x be the solution to Ax = bLet y be the solution to Ay = cThen a simple analysis shows that

Errors in the data (b) are magnified by cond(A)

Likewise for errors in A

cond( )x y b c

Ax c