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

n = norm(v) returns the Euclidean norm of vector v. This norm is also called the 2-norm, vector magnitude, or Euclidean length.

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

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

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

• If p = 1, then n is the maximum absolute column sum of the matrix.

• If p = 2, then n is approximately max(svd(X)). This value is equivalent to norm(X).

• If p = Inf, then n is the maximum absolute row sum of the matrix.

n = norm(X,"fro") returns the Frobenius norm of matrix or array X.

Examples

Vector Magnitude

Create a vector and calculate the magnitude.

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

n = 
3.7417

1-Norm of Vector

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

v = [-2 3 -1];
n = norm(v,1)

n = 
6

Euclidean Distance Between Two Points

Calculate the distance between two points as the norm of the difference between the vector elements.

Create two vectors representing the (x,y) coordinates for two points on the Euclidean plane.

a = [0 3];
b = [-2 1];

Use norm to calculate the distance between the points.

d = norm(b-a)

d = 
2.8284

Geometrically, the distance between the points is equal to the magnitude of the vector that extends from one point to the other.

$$\begin{array}{l}a=0\underset{}{\overset{ˆ}{i}}+3\underset{}{\overset{ˆ}{j}}\\ b=-2\underset{}{\overset{ˆ}{i}}+1\underset{}{\overset{ˆ}{j}}\\ \\ \begin{array}{rl}{d}_{(a,b)}& =||b-a||\\ & =\sqrt{(-2-0{)}^2+(1-3{)}^2}\\ & =\sqrt{8}\end{array}\end{array}$$

2-Norm of Matrix

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

Frobenius Norm of N-D Array

Calculate the Frobenius norm of a 4-D array X, which is equivalent to the 2-norm of the column vector X(:).

X = rand(3,4,4,3);
n = norm(X,"fro")

n = 
7.1247

The Frobenius norm is also useful for sparse matrices because norm(X,2) does not support sparse X.

Input Arguments

v — Input vector

vector

Input vector.

Data Types: single | double
Complex Number Support: Yes

X — Input array

matrix | array

Input array, specified as a matrix or array. For most norm types, X must be a matrix. However, for Frobenius norm calculations, X can be an array.

Data Types: single | double
Complex Number Support: Yes

p — Norm type

2 (default) | positive real scalar | Inf | -Inf

Norm type, specified as 2 (default), a positive real 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.

p	Matrix	Vector
1	max(sum(abs(X)))	sum(abs(v))
2	max(svd(X))	sum(abs(v).^2)^(1/2)
Positive, real-valued numeric scalar	—	sum(abs(v).^p)^(1/p)
Inf	max(sum(abs(X')))	max(abs(v))
-Inf	—	min(abs(v))

Output Arguments

n — Norm value

scalar

Norm value, 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.

• The norm of an empty matrix is zero.

More About

Euclidean Norm

The Euclidean norm (also called the vector magnitude, Euclidean length, or 2-norm) of a vector v with N elements is defined by

$$\Vert v\Vert =\sqrt{{\displaystyle \sum _{k=1}^N{\left|v_k\right|}^2}}.$$

General Vector Norm

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

$${\Vert v\Vert }_p={\left[{\displaystyle \sum _{k=1}^N{\left|v_k\right|}^p}\right]}^{1/p},$$

where p is any positive real value, Inf, or -Inf.

• If p = 1, then the resulting 1-norm is the sum of the absolute values of the vector elements.

• If p = 2, then the resulting 2-norm gives the vector magnitude or Euclidean length of the vector.

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

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

Maximum Absolute Column Sum

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

$${\Vert X\Vert }_1=\underset{1\le j\le n}{\max }\left({\displaystyle \sum _{i=1}^m\left|a_{ij}\right|}\right).$$

Maximum Absolute Row Sum

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

$${\Vert X\Vert }_{\infty }=\underset{1\le i\le m}{\max }\left({\displaystyle \sum _{j=1}^n\left|a_{ij}\right|}\right).$$

Frobenius Norm

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

$${\Vert X\Vert }_F=\sqrt{{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\left|a_{ij}\right|}^2}}}=\sqrt{\text{trace}\left(X^{†}X\right)}.$$

This definition also extends naturally to arrays with more than two dimensions. For example, if X is an N-D array of size m-by-n-by-p-by-...-by-q, then the Frobenius norm is

$${\Vert X\Vert }_F=\sqrt{{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^p\mathrm{...}{\displaystyle \sum _{w=1}^q{\left|a_{ijk\mathrm{...}w}\right|}^2}}}}}.$$

Tips

• Use vecnorm to treat a matrix or array as a collection of vectors and calculate the norm along a specified dimension. For example, vecnorm can calculate the norm of each column in a matrix.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

The norm function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Code generation does not support sparse matrix inputs for this function.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The norm function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2022a: Frobenius norm supports N-D arrays

Norm calculations of the form norm(X,"fro") support N-D array inputs.