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Student[LinearAlgebra]

  

Norm

  

compute the p-norm of a Matrix or Vector

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Norm(A, p, options)

Parameters

A

-

Matrix or Vector

p

-

(optional) non-negative number, infinity, Euclidean, or Frobenius; norm selector that is dependent upon A

options

-

(optional) parameters; for a complete list, see LinearAlgebra[Norm]

Description

• 

The Norm(A) command computes the Euclidean (2)-norm of A.

  

Note: The default norm in the top-level LinearAlgebra package is the infinity norm, as that norm is faster to compute for Matrices.

  

The allowable values for the norm-selector parameter, p, depend on whether A is a Vector or a Matrix.

  

 

  

Vector Norms

• 

If V is a Vector and p is included in the calling sequence, p must be one of a non-negative number, infinity, Frobenius, or Euclidean.

  

The p-norm of a Vector V when 1<=p<infinity is addVip&comma;i&equals;1..DimensionV1p.

  

The infinity-norm of  Vector V is maxseqVi&comma;i&equals;1..DimensionV.

  

Maple implements Vector norms for all 0<=p<=infinity.  For 0<p<1 the final pth root computation is not done, that is, the calculation is addVip&comma;i&equals;1..DimensionV. This defines a metric on Rn, but the pth root is not a norm and the form computed by Norm in such cases is more useful.  The limiting case of p&equals;0 returns the number of nonzero elements of V (this is a floating-point number  if p or any element of V is a floating-point number).

  

For Vectors, the 2-norm can also be specified as either Euclidean or  Frobenius.

  

 

  

Matrix Norms

• 

If A is a Matrix and p is included in the calling sequence, p must be one of 1, 2, infinity, Frobenius, or Euclidean.

  

The p-norm of a Matrix A is max(Norm(A . V, p)), where the maximum is calculated over all Vectors V with Norm(V, p) = 1.  Maple implements only Norm(A, p) for p=1,2, and the special case p&equals;Frobenius (which is not actually a Matrix norm; the Matrix A is treated as a "folded up" Vector). These norms are defined as follows.

  

Norm(A, 1) = max(seq(Norm(A[1..-1, j], 1), j = 1 .. ColumnDimension(A)))

  

Norm(A, infinity) = max(seq(Norm(A[i, 1..-1], 1), i = 1 .. RowDimension(A)))

  

Norm(A, 2) = sqrt(max(seq(Eigenvalues(A . A^%T)[i], i = 1 .. RowDimension(A))))

  

Norm(A, Frobenius) = sqrt(add(add((A[i,j]^2), j = 1 .. ColumnDimension(A)), i = 1 .. RowDimension(A)))

  

For Matrices, the 2-norm can also be specified as Euclidean.

Examples

withStudent&lsqb;LinearAlgebra&rsqb;&colon;

A1&comma;1&comma;0&verbar;0&comma;1&comma;1&verbar;1&comma;0&comma;1

A:=101110011

(1)

NormA&comma;2

3

(2)

B10&comma;0&comma;0&verbar;0&comma;9&comma;12&verbar;2&comma;4&comma;1

B:=10020940121

(3)

NormB&comma;1

10

(4)

h3&verbar;4

h:=34

(5)

hNormh&comma;1

3747

(6)

va&comma;b&comma;c

v:=abc

(7)

Normv&comma;&infin;

maxa&comma;b&comma;c

(8)

See Also

LinearAlgebra[Norm]

Student[LinearAlgebra]

Student[LinearAlgebra][Normalize]

Student[LinearAlgebra][Operators]

 


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