# Documentation

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# norm

Vector and matrix norms

## Syntax

n = norm(v)
n = norm(v,p)
n = norm(X)
n = norm(X,p)
n = norm(X,'fro')

## Description

example

n = norm(v) returns the 2-norm or Euclidean norm of vector v.

example

n = norm(v,p) returns the generalized vector p-norm.

example

n = norm(X) returns the 2-norm or maximum singular value of matrix X, which is approximately max(svd(X)).

example

n = norm(X,p) returns the p-norm of matrix X, where p is 1, 2, or Inf:

example

n = norm(X,'fro') returns the Frobenius norm of matrix X.

## Examples

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Calculate the 2-norm of a vector corresponding to the point (-2,3,-1) in 3-D space. The 2-norm is equal to the Euclidean length of the vector.

X = [-2 3 -1];
n = norm(X)
n = 3.7417

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

n = norm(X,1)
n = 6

Calculate the 2-norm of a matrix, which is the largest singular value.

X = [2 0 1;-1 1 0;-3 3 0];
n = norm(X)
n = 4.7234

Use 'fro' to calculate the Frobenius norm of a sparse matrix, which calculates the 2-norm of the column vector, S(:).

S = sparse(1:25,1:25,1);
n = norm(S,'fro')
n = 5

## Input Arguments

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Input vector.

Data Types: single | double
Complex Number Support: Yes

Input matrix.

Data Types: single | double
Complex Number Support: Yes

Norm type, specified as 2 (default), a different positive integer scalar, Inf, or -Inf. The valid values of p and what they return depend on whether the first input to norm is a matrix or vector, as shown in the table.

 Note:   This table does not reflect the actual algorithms used in calculations.
pMatrixVector
1max(sum(abs(X)))sum(abs(X))
2max(svd(X))sum(abs(X).^2)^(1/2)
Positive, real-valued numeric psum(abs(X).^p)^(1/p)
Infmax(sum(abs(X')))max(abs(X))
-Infmin(abs(X))

## Output Arguments

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Matrix or vector norm, returned as a scalar. The norm gives a measure of the magnitude of the elements. By convention, norm returns NaN if the input contains NaN values.

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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, Inf, or -Inf.

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

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

### Maximum Absolute Column Sum

The maximum absolute column sum of an m-by-n matrix X (with m,n >= 2) is defined by

${‖X‖}_{1}=\underset{1\le j\le n}{\mathrm{max}}\left(\sum _{i=1}^{m}|{a}_{ij}|\right).$

### Maximum Absolute Row Sum

The maximum absolute row sum of an m-by-n matrix X (with m,n >= 2) is defined by

${‖X‖}_{\infty }=\underset{1\le i\le m}{\mathrm{max}}\left(\sum _{j=1}^{n}|{a}_{ij}|\right)\text{\hspace{0.17em}}.$

### Frobenius Norm

The Frobenius norm of an m-by-n matrix X (with m,n >= 2) is defined by

${‖X‖}_{F}=\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{|{a}_{ij}|}^{2}}=\sqrt{\text{trace}\left({X}^{†}X\right)}\text{\hspace{0.17em}}.$