vector norm

Vector Norm

Given an -dimensional vector

(1)

a general vector norm , sometimes written with a double bar as , is a nonnegative norm defined such that

1.  when  and  iff .

2.  for any scalar .

3. .

In this work, a single bar is used to denote a vector norm, absolute value, or complex modulus, while a double bar is reserved for denoting a matrix norm.

The vector norm  for , 2, ... is defined as

(2)

The -norm of vector  is implemented as Norm[v, p], with the 2-norm being returned by Norm[v].

The special case  is defined as

(3)

The most commonly encountered vector norm (often simply called "the norm" of a vector, or sometimes the magnitude of a vector) is the L2-norm, given by

(4)

This and other types of vector norms are summarized in the following table, together with the value of the norm for the example vector .

name	symbol	value	approx.
-norm		6	6.000
-norm			3.742
-norm			3.302
-norm			3.146
-norm		3	3.000