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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
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Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Applications  
Properties & Relations  
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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

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Basic Examples  (1)

Scope  (5)

Symbolic vectors:

Use an arbitrary norm function:

v is a complex‐valued vector:

Normalize using exact arithmetic:

Use machine arithmetic:

Use 24‐digit 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_2025_normalize, author="Wolfram Research", title="{Normalize}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Normalize.html}", note=[Accessed: 08-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_normalize, organization={Wolfram Research}, title={Normalize}, year={2007}, url={https://reference.wolfram.com/language/ref/Normalize.html}, note=[Accessed: 08-August-2025]}