Normalize

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`Normalize`

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Normalize[v]

gives the normalized form of a vector v.

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Normalize[z]

gives the normalized form of a complex number z.

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Normalize[expr,f]

normalizes with respect to the norm function f.

Details

Normalize[v] is effectively v/Norm[v], except that zero vectors are returned unchanged.
Except in the case of zero vectors, Normalize[v] returns the unit vector in the direction of v.
For a complex number z, Normalize[z] returns z/Abs[z], except that Normalize[0] gives 0.
Normalize[expr,f] is effectively expr/f[expr], except when there are zeros in f[expr].

Examples

open allclose all

Basic Examples (1)Summary of the most common use cases

In[1]:=1

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https://wolfram.com/xid/0binefxg-wwmef

Out[1]=1

Scope (5)Survey of the scope of standard use cases

Symbolic vectors:

In[1]:=1

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https://wolfram.com/xid/0binefxg-lvafnn

Out[1]=1

Use an arbitrary norm function:

In[1]:=1

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https://wolfram.com/xid/0binefxg-iwgj6a

Out[1]=1

v is a complex‐valued vector:

In[1]:=1

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https://wolfram.com/xid/0binefxg-yvefc

Normalize using exact arithmetic:

In[2]:=2

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https://wolfram.com/xid/0binefxg-l8ce20

Out[2]=2

Use machine arithmetic:

In[3]:=3

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https://wolfram.com/xid/0binefxg-di4n8p

Out[3]=3

Use 24‐digit precision arithmetic:

In[4]:=4

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https://wolfram.com/xid/0binefxg-o499d5

Out[4]=4

Normalize a sparse vector:

In[1]:=1

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https://wolfram.com/xid/0binefxg-cz3qh2

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0binefxg-c28yhw

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0binefxg-b8uret

Out[3]=3

Normalize a TimeSeries:

In[1]:=1

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https://wolfram.com/xid/0binefxg-397g5y

In[2]:=2

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https://wolfram.com/xid/0binefxg-8gd40a

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0binefxg-1hdxcf

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0binefxg-yawamn

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0binefxg-bfacom

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0binefxg-6b5bfc

Out[6]=6

Generalizations & Extensions (2)Generalized and extended use cases

Normalize a matrix by explicitly specifying a norm function:

In[1]:=1

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https://wolfram.com/xid/0binefxg-cir1dn

Out[1]=1

Normalize a polynomial with respect to integration over the interval to :

In[1]:=1

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https://wolfram.com/xid/0binefxg-b3969g

Out[1]=1

Applications (1)Sample problems that can be solved with this function

m is a symmetric matrix with distinct eigenvalues:

In[1]:=1

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https://wolfram.com/xid/0binefxg-xdjek

Out[1]=1

Power method to find the eigenvector associated with the largest eigenvalue:

In[2]:=2

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https://wolfram.com/xid/0binefxg-1b3ud

Out[2]=2

This is consistent (up to sign) with what Eigenvectors gives:

In[3]:=3

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https://wolfram.com/xid/0binefxg-h4js08

Out[3]=3

The eigenvalue can be found with Norm:

In[4]:=4

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https://wolfram.com/xid/0binefxg-hqvb10

Out[4]=4

Properties & Relations (1)Properties of the function, and connections to other functions

v is a random vector:

In[5]:=5

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https://wolfram.com/xid/0binefxg-bjrzf

Out[1]=1

u is the normalization of v:

In[2]:=2

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https://wolfram.com/xid/0binefxg-jhgvl

Out[2]=2

u is a unit vector in the direction of v:

In[3]:=3

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https://wolfram.com/xid/0binefxg-x2maf

Out[3]=3

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