norm

Norm of vector or matrix

Syntax

norm(v)

norm(v,p)

norm(A)

norm(A,P)

Description

norm(v) returns the 2-norm of vector v.

norm(v,p) returns the p-norm of vector v.

norm(A) returns the 2-norm of matrix A. Because symbolic variables are assumed to be complex by default, the norm can contain unresolved calls to conj and abs.

norm(A,P) returns the P-norm of matrix A.

Examples

Compute 2-Norm of Matrix

Compute the 2-norm of the inverse of the 3-by-3 magic square A:

A = inv(sym(magic(3)))
norm2 = norm(A)

A =
[  53/360, -13/90,  23/360]
[ -11/180,   1/45,  19/180]
[  -7/360,  17/90, -37/360]
 
norm2 =
3^(1/2)/6

Use vpa to approximate the result with 20-digit accuracy:

vpa(norm2, 20)

ans =
0.28867513459481288225

Effect of Assumptions on Norm

Compute the norm of [x y] and simplify the result. Because symbolic scalar variables are assumed to be complex by default, the calls to abs do not simplify.

syms x y
simplify(norm([x y]))

ans =
(abs(x)^2 + abs(y)^2)^(1/2)

Assume x and y are real, and repeat the calculation. Now, the result is simplified.

assume([x y],'real')
simplify(norm([x y]))

ans =
(x^2 + y^2)^(1/2)

Remove assumptions on x for further calculations. For details, see Use Assumptions on Symbolic Variables.

assume(x,'clear')

Compute Different Types of Norms of Matrix

Compute the 1-norm, Frobenius norm, and infinity norm of the inverse of the 3-by-3 magic square A:

A = inv(sym(magic(3)))
norm1 = norm(A, 1)
normf = norm(A, 'fro')
normi = norm(A, inf)

A =
[  53/360, -13/90,  23/360]
[ -11/180,   1/45,  19/180]
[  -7/360,  17/90, -37/360]
 
norm1 =
16/45
 
normf =
391^(1/2)/60
 
normi =
16/45

Use vpa to approximate these results to 20-digit accuracy:

vpa(norm1, 20)
vpa(normf, 20)
vpa(normi, 20)

ans =
0.35555555555555555556
 
ans =
0.32956199888808647519
 
ans =
0.35555555555555555556

Compute Different Types of Norms of Vector

Compute the 1-norm, 2-norm, and 3-norm of the column vector V = [Vx; Vy; Vz]:

syms Vx Vy Vz
V = [Vx; Vy; Vz];
norm1 = norm(V, 1)
norm2 = norm(V)
norm3 = norm(V, 3)

norm1 =
abs(Vx) + abs(Vy) + abs(Vz)
 
norm2 =
(abs(Vx)^2 + abs(Vy)^2 + abs(Vz)^2)^(1/2)
 
norm3 =
(abs(Vx)^3 + abs(Vy)^3 + abs(Vz)^3)^(1/3)

Compute the infinity norm, negative infinity norm, and Frobenius norm of V:

normi = norm(V, inf)
normni = norm(V, -inf)
normf = norm(V, 'fro')

normi =
max(abs(Vx), abs(Vy), abs(Vz))
 
normni =
min(abs(Vx), abs(Vy), abs(Vz))
 
normf =
(abs(Vx)^2 + abs(Vy)^2 + abs(Vz)^2)^(1/2)

Input Arguments

v — Input vector

vector of symbolic scalar variables | symbolic matrix variable

Input vector, specified as a vector of symbolic scalar variables, or a symbolic matrix variable (since R2021a) that represents a vector.

p — Input

2 (default) | 1 | positive integer scalar | Inf | -Inf | 'fro'

• norm(v,p) is computed as sum(abs(v).^p)^(1/p) for 1<=p<inf.

• norm(v) computes the 2-norm of V.

• norm(v,Inf) is computed as max(abs(V)).

• norm(v,-Inf) is computed as min(abs(V)).

A — Input matrix

matrix of symbolic scalar variables | symbolic matrix variable

Input matrix, specified as a matrix of symbolic scalar variables, or a symbolic matrix variable (since R2021a) that represents a matrix.

P — Input

2 (default) | 1 | Inf | 'fro'

One of these values 1, 2, Inf, or 'fro'.

• norm(A,1) returns the 1-norm of A.

• norm(A,2) or norm(A) returns the 2-norm of A.

• norm(A,Inf) returns the infinity norm of A.

• norm(A,'fro') returns the Frobenius norm of A.

More About

1-Norm of a Matrix

The 1-norm of an m-by-n matrix A is defined as follows:

$${\Vert A\Vert }_1=\underset{j}{\max }\left({\displaystyle \sum _{i=1}^m\left|A_{ij}\right|}\right),\text{ where }j=1\dots n$$

2-Norm of a Matrix

The 2-norm of an m-by-n matrix A is defined as follows:

$${\Vert A\Vert }_2=\sqrt{{\text{max eigenvalue of A}}^{H}A}$$

The 2-norm is also called the spectral norm of a matrix.

Frobenius Norm of a Matrix

The Frobenius norm of an m-by-n matrix A is defined as follows:

$${\Vert A\Vert }_F=\sqrt{{\displaystyle \sum _{i=1}^m\left({\displaystyle \sum _{j=1}^n{\left|A_{ij}\right|}^2}\right)}}$$

Infinity Norm of a Matrix

The infinity norm of an m-by-n matrix A is defined as follows:

$${\Vert A\Vert }_{\infty }=\max \left({\displaystyle \sum _{j=1}^n\left|A_{1j}\right|,}{\displaystyle \sum _{j=1}^n\left|A_{2j}\right|,}\dots ,{\displaystyle \sum _{j=1}^n\left|A_{mj}\right|}\right)$$

P-Norm of a Vector

The P-norm of a 1-by-n or n-by-1 vector V is defined as follows:

$${\Vert V\Vert }_P={\left({\displaystyle \sum _{i=1}^n{\left|V_i\right|}^P}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$P$}\right.}$$

Here n must be an integer greater than 1.

Frobenius Norm of a Vector

The Frobenius norm of a 1-by-n or n-by-1 vector V is defined as follows:

$${\Vert V\Vert }_F=\sqrt{{\displaystyle \sum _{i=1}^n{\left|V_i\right|}^2}}$$

The Frobenius norm of a vector coincides with its 2-norm.

Infinity and Negative Infinity Norm of a Vector

The infinity norm of a 1-by-n or n-by-1 vector V is defined as follows:

$${\Vert V\Vert }_{\infty }=\max \left(\left|V_i\right|\right),\text{ where }i=1\dots n$$

The negative infinity norm of a 1-by-n or n-by-1 vector V is defined as follows:

$${\Vert V\Vert }_{-\infty }=\min \left(\left|V_i\right|\right),\text{ where }i=1\dots n$$

Tips

• Calling norm for a numeric matrix that is not a symbolic object invokes the MATLAB^® norm function.