# norm

Norm of symbolic vector or matrix

## Description

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

example

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

example

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.

example

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

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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 = $0.28867513459481288225$

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 = $\sqrt{{|x|}^{2}+{|y|}^{2}}$

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 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 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 = $|\mathrm{Vx}|+|\mathrm{Vy}|+|\mathrm{Vz}|$
norm2 = norm(V)
norm2 = $\sqrt{{|\mathrm{Vx}|}^{2}+{|\mathrm{Vy}|}^{2}+{|\mathrm{Vz}|}^{2}}$
norm3 = norm(V,3)
norm3 = ${\left({|\mathrm{Vx}|}^{3}+{|\mathrm{Vy}|}^{3}+{|\mathrm{Vz}|}^{3}\right)}^{1/3}$

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

normi = norm(V,Inf)
normi = $\mathrm{max}\left(|\mathrm{Vx}|,|\mathrm{Vy}|,|\mathrm{Vz}|\right)$
normni = norm(V,-Inf)
normni = $\mathrm{min}\left(|\mathrm{Vx}|,|\mathrm{Vy}|,|\mathrm{Vz}|\right)$
normf = norm(V,"fro")
normf = $\sqrt{{|\mathrm{Vx}|}^{2}+{|\mathrm{Vy}|}^{2}+{|\mathrm{Vz}|}^{2}}$

## Input Arguments

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

• 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)).

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

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.

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

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### 1-Norm of a Matrix

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

### 2-Norm of a Matrix

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

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:

${‖A‖}_{\infty }=\mathrm{max}\left(\sum _{j=1}^{n}|{A}_{1j}|,\text{\hspace{0.17em}}\sum _{j=1}^{n}|{A}_{2j}|,\dots ,\sum _{j=1}^{n}|{A}_{mj}|\right)$

### Frobenius Norm of a Matrix and Multidimensional Array

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

${‖A‖}_{F}=\sqrt{\sum _{i=1}^{m}\left(\sum _{j=1}^{n}{|{A}_{ij}|}^{2}\right)}$

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

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

### P-Norm of a Vector

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

${‖V‖}_{P}={\left(\sum _{i=1}^{n}{|{V}_{i}|}^{P}\right)}^{1}{P}}$

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:

${‖V‖}_{F}=\sqrt{\sum _{i=1}^{n}{|{V}_{i}|}^{2}}$

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:

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

## Tips

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

## Version History

Introduced in R2012b

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