norm
Norm of symbolic vector or matrix
Description
returns the
n
= norm(v
)2
-norm or the magnitude of symbolic vector
v
.
returns the
n
= norm(A
)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
.
returns the Frobenius norm of symbolic multidimensional array
n
= norm(X
,"fro")X
.
Examples
Magnitude of Symbolic Vector
Create a symbolic vector and calculate the magnitude.
syms x y z r = [x y z]
r =
n = norm(r)
n =
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 =
norm2 = norm(A)
norm2 =
Use vpa
to approximate the result with 20-digit accuracy.
norm2_vpa = vpa(norm2,20)
norm2_vpa =
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 =
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 =
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 =
norm1 = norm(A,1)
norm1 =
normf = norm(A,"fro")
normf =
normi = norm(A,Inf)
normi =
Use vpa
to approximate these results to 20-digit accuracy.
norm1_vpa = vpa(norm1,20)
norm1_vpa =
normf_vpa = vpa(normf,20)
normf_vpa =
normi_vpa = vpa(normi,20)
normi_vpa =
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 =
norm2 = norm(V)
norm2 =
norm3 = norm(V,3)
norm3 =
Compute the infinity norm, negative infinity norm, and Frobenius norm of V
.
normi = norm(V,Inf)
normi =
normni = norm(V,-Inf)
normni =
normf = norm(V,"fro")
normf =
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 assum(abs(v).^p)^(1/p)
for1<=p<Inf
.norm(v)
computes the2
-norm ofV
.norm(v,Inf)
is computed asmax(abs(V))
.norm(v,-Inf)
is computed asmin(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 the1
-norm ofA
.norm(A,2)
ornorm(A)
returns the2
-norm ofA
.norm(A,Inf)
returns the infinity norm ofA
.norm(A,"fro")
returns the Frobenius norm ofA
.
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:
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:
Frobenius Norm of a Matrix and Multidimensional Array
The Frobenius norm of an m-by-n matrix A is defined as follows:
The Frobenius norm of an l-by-m-by-n multidimensional array X is defined as follows:
P-Norm of a Vector
The P
-norm of a 1-by-n or
n-by-1 vector V is defined as follows:
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:
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 R2012bR2022a: 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
.
See Also
cond
| equationsToMatrix
| inv
| linsolve
| rank
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