Documentation

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 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 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.

X = [-2 3 -1];
n = norm(X,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.

$\begin{array}{l} a = 0 i_{}^{ˆ} + 3 j_{}^{ˆ} \\ b = - 2 i_{}^{ˆ} + 1 j_{}^{ˆ} \\ \begin{aligned} d_{(a, b)} & = | | b - a | | \\ = \sqrt{(- 2 - 0)^{2} + (1 - 3)^{2}} \\ = \sqrt{8} \end{aligned} \end{array}$

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 Sparse Matrix

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

`v` — Input vector
vector

Input vector.

Data Types: single | double
Complex Number Support: Yes

`X` — Input matrix
matrix

Input matrix.

Data Types: single | double
Complex Number Support: Yes

`p` — Norm type
2 (default) | positive integer scalar | `Inf` | `-Inf`

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.

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

Output Arguments

`n` — Matrix or vector norm
scalar

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.

More About