A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

Linear Algebra? 23

Since this is the most commonly used norm, when no sub-

script appears next to the norm notation, it usually means

the l

2

-norm.

• The l

1

-norm, which is the sum of absolute values:

kuk

1

= |u

1

| + |u

2

| + ··· + |u

n

|;

• The l

∞

-norm, which is also known as the max norm or in-

finity norm:

kuk

∞

= max

i

|u

i

|.

Figure 2.12: Unit circles for various norms.

Next, we briefly mention how matrix norms are defined. Espe-

cially important are induced matrix norms, which are defined as

follows:

kAk = max

u6=0

kAuk

kuk

,

where u is a vector. For example, the 2-norm of a matrix is de-

fined as kAk

2

= max

u6=0

kAuk

2

kuk

2

, where kAuk

2

and kuk

2

are the

previously defined Euclidean ...