# Documentation

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# vecnorm

Vector-wise norm

## Syntax

N = vecnorm(A)
N = vecnorm(A,p)
N = vecnorm(A,p,dim)

## Description

example

N = vecnorm(A) returns the 2-norm or Euclidean norm of A:

• If A is a vector, then vecnorm returns the norm of the vector.

• If A is a matrix, then vecnorm returns the norm of each column.

• If A is a multidimensional array, then vecnorm returns the norm along the first array dimension whose size does not equal 1.

example

N = vecnorm(A,p) calculates the generalized vector p-norm.

example

N = vecnorm(A,p,dim) operates along dimension dim. The size of this dimension reduces to 1 while the sizes of all other dimensions remain the same.

## Examples

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Calculate the 2-norm of a vector corresponding to the point (2,2,2) in 3-D space. The 2-norm is equal to the Euclidean length of the vector, .

x = [2 2 2];
n = vecnorm(x)
n =

3.4641

Calculate the 1-norm of the vector, which is the sum of the element magnitudes.

n = vecnorm(x,1)
n =

6

Calculate the 2-norm of the columns of a matrix.

A = [2 0 1;-1 1 0;-3 3 0]
A =

2     0     1
-1     1     0
-3     3     0

n = vecnorm(A)
n =

3.7417    3.1623    1.0000

As an alternative, you can use the norm function to calculate the 2-norm of the entire matrix.

## Input Arguments

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Input array, specified as a vector, matrix, or multidimensional array. By convention, vecnorm returns NaN values if the vector being operated on contains a NaN value.

Data Types: single | double
Complex Number Support: Yes

Norm type, specified as 2 (default), a positive scalar, or Inf.

Dimension to operate along, specified as a positive integer scalar. If you do not specify a value, then the default is the first array dimension whose size does not equal 1.

Dimension dim indicates the dimension whose length reduces to 1. In other words, size(N,dim) is 1, while the sizes of all other dimensions remain the same.

Consider a two-dimensional input array, A:

• vecnorm(A,p,1) calculates the norm of each column.

• vecnorm(A,p,2) calculates the norm of each row.

vecnorm returns abs(A) when dim is greater than ndims(A) or when size(A,dim) is 1.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

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### Euclidean Norm

The Euclidean norm (or 2-norm) of a vector v that has N elements is defined by

$‖v‖=\sqrt{\sum _{k=1}^{N}{|{v}_{k}|}^{2}}\text{\hspace{0.17em}}.$

### General Vector Norm

The general definition for the vector p-norm of a vector v that has N elements is

${‖v‖}_{p}={\left[\sum _{k=1}^{N}{|{v}_{k}|}^{p}\right]}^{\text{\hspace{0.17em}}1/p}\text{\hspace{0.17em}},$

where p is any positive real value or Inf.

If p = Inf, then ${‖v‖}_{\infty }={\mathrm{max}}_{i}\left(|v\left(i\right)|\right)$.