norm

Norm of symbolic vector or matrix

Syntax

n = norm(v)

n = norm(v,p)

n = norm(A)

n = norm(A,P)

n = norm(X,"fro")

Description

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

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

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

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

n = norm(X,"fro") returns the Frobenius norm of symbolic multidimensional array X.

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)))

A = 
$$\left(\begin{array}{ccc}\frac{53}{360}& -\frac{13}{90}& \frac{23}{360}\\ -\frac{11}{180}& \frac{1}{45}& \frac{19}{180}\\ -\frac{7}{360}& \frac{17}{90}& -\frac{37}{360}\end{array}\right)$$

norm2 = norm(A)

norm2 = 
$$\frac{\sqrt{3}}{6}$$

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

norm2_vpa = vpa(norm2,20)

norm2_vpa = $$0.28867513459481288225$$

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
n = simplify(norm([x y]))

n = $$\sqrt{{\left|x\right|}^2+{\left|y\right|}^2}$$

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

assume([x y],"real")
n = simplify(norm([x y]))

n = $$\sqrt{x^2+y^2}$$

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)))

A = 
$$\left(\begin{array}{ccc}\frac{53}{360}& -\frac{13}{90}& \frac{23}{360}\\ -\frac{11}{180}& \frac{1}{45}& \frac{19}{180}\\ -\frac{7}{360}& \frac{17}{90}& -\frac{37}{360}\end{array}\right)$$

norm1 = norm(A,1)

norm1 = 
$$\frac{16}{45}$$

normf = norm(A,"fro")

normf = 
$$\frac{\sqrt{391}}{60}$$

normi = norm(A,Inf)

normi = 
$$\frac{16}{45}$$

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

norm1_vpa = vpa(norm1,20)

norm1_vpa = $$0.35555555555555555556$$

normf_vpa = vpa(normf,20)

normf_vpa = $$0.32956199888808647519$$

normi_vpa = vpa(normi,20)

normi_vpa = $$0.35555555555555555556$$

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)

norm1 = $$\left|\mathrm{Vx}\right|+\left|\mathrm{Vy}\right|+\left|\mathrm{Vz}\right|$$

norm2 = norm(V)

norm2 = $$\sqrt{{\left|\mathrm{Vx}\right|}^2+{\left|\mathrm{Vy}\right|}^2+{\left|\mathrm{Vz}\right|}^2}$$

norm3 = norm(V,3)

norm3 = $${\left({\left|\mathrm{Vx}\right|}^3+{\left|\mathrm{Vy}\right|}^3+{\left|\mathrm{Vz}\right|}^3\right)}^{1/3}$$

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

normi = norm(V,Inf)

normi = $$\max \left(\left|\mathrm{Vx}\right|,\left|\mathrm{Vy}\right|,\left|\mathrm{Vz}\right|\right)$$

normni = norm(V,-Inf)

normni = $$\min \left(\left|\mathrm{Vx}\right|,\left|\mathrm{Vy}\right|,\left|\mathrm{Vz}\right|\right)$$

normf = norm(V,"fro")

normf = $$\sqrt{{\left|\mathrm{Vx}\right|}^2+{\left|\mathrm{Vy}\right|}^2+{\left|\mathrm{Vz}\right|}^2}$$

Input Arguments

v — Input vector

vector of symbolic scalar variables | symbolic matrix variable | symbolic function | symbolic matrix function

Input vector, specified as a vector of symbolic scalar variables, symbolic matrix variable, function, or matrix function 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 | symbolic function | symbolic matrix function

Input matrix, specified as a matrix of symbolic scalar variables, symbolic matrix variable, function, or matrix function 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.

X — Input array

multidimensional array of symbolic scalar variables

Input array, specified as a multidimensional array of symbolic scalar variables.

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.

Infinity Norm of a Matrix

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

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

Frobenius Norm of a Matrix and Multidimensional Array

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

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

The Frobenius norm of an l-by-m-by-n multidimensional array X is defined as follows:

$${\Vert X\Vert }_F=\sqrt{{\displaystyle \sum _{i=1}^l\left({\displaystyle \sum _{j=1}^m\left({\displaystyle \sum _{k=1}^n{\left|X_{ijk}\right|}^2}\right)}\right)}}$$

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.

Version History

Introduced in R2012b

R2022a: Compute Frobenius norm of symbolic array

The norm function accepts a symbolic multidimensional array as its input argument. Use the syntax norm(X,"fro") to return the Frobenius norm of a symbolic array X.

R2022a: Compute norm of symbolic matrix functions

The norm function accepts an input argument of type symfunmatrix.

R2021a: Compute norm of symbolic matrix variables

The norm function accepts an input argument of type symmatrix.