# Normalize

Normalize[v]

gives the normalized form of a vector v.

Normalize[z]

gives the normalized form of a complex number z.

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

## Scope(5)

Symbolic vectors:

Use an arbitrary norm function:

v is a complexvalued vector:

Normalize using exact arithmetic:

Use machine arithmetic:

Use 24digit precision arithmetic:

Normalize a sparse vector:

Normalize a TimeSeries:

## Generalizations & Extensions(2)

Normalize a matrix by explicitly specifying a norm function:

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

## Applications(1)

m is a symmetric matrix with distinct eigenvalues:

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

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

The eigenvalue can be found with Norm:

## Properties & Relations(1)

v is a random vector:

u is the normalization of v:

u is a unit vector in the direction of v:

Wolfram Research (2007), Normalize, Wolfram Language function, https://reference.wolfram.com/language/ref/Normalize.html.

#### Text

Wolfram Research (2007), Normalize, Wolfram Language function, https://reference.wolfram.com/language/ref/Normalize.html.

#### CMS

Wolfram Language. 2007. "Normalize." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Normalize.html.

#### APA

Wolfram Language. (2007). Normalize. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Normalize.html

#### BibTeX

@misc{reference.wolfram_2023_normalize, author="Wolfram Research", title="{Normalize}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Normalize.html}", note=[Accessed: 01-December-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_normalize, organization={Wolfram Research}, title={Normalize}, year={2007}, url={https://reference.wolfram.com/language/ref/Normalize.html}, note=[Accessed: 01-December-2023 ]}