Norm - Maple Help

Student[LinearAlgebra]

 Norm
 compute the p-norm of a Matrix or Vector

 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 $1\le p<\mathrm{\infty }$ is ${\mathrm{add}\left({\left|{V}_{i}\right|}^{p},i=1..\mathrm{Dimension}\left(V\right)\right)}^{\frac{1}{p}}$.
 The infinity-norm of  Vector V is $\mathrm{max}\left(\mathrm{seq}\left(\left|{V}_{i}\right|,i=1..\mathrm{Dimension}\left(V\right)\right)\right)$.
 Maple implements Vector norms for all $0\le p\le \mathrm{\infty }$.  For $0 the final pth root computation is not done, that is, the calculation is $\mathrm{add}\left({\left|{V}_{i}\right|}^{p},i=1..\mathrm{Dimension}\left(V\right)\right)$. This defines a metric on ${R}^{n}$, but the pth root is not a norm and the form computed by Norm in such cases is more useful.  The limiting case of $p=0$ 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 $p=1,2,\mathrm{\infty }$ and the special case $p=\mathrm{Frobenius}$ (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

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$
 > $A≔⟨⟨1,-1,0⟩|⟨0,1,1⟩|⟨1,0,1⟩⟩$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {0}& {1}\\ {-1}& {1}& {0}\\ {0}& {1}& {1}\end{array}\right]$ (1)
 > $\mathrm{Norm}\left(A,2\right)$
 $\sqrt{{3}}$ (2)
 > $B≔⟨⟨10,0,0⟩|⟨0,9,\frac{1}{2}⟩|⟨2,4,1⟩⟩$
 ${B}{≔}\left[\begin{array}{ccc}{10}& {0}& {2}\\ {0}& {9}& {4}\\ {0}& \frac{{1}}{{2}}& {1}\end{array}\right]$ (3)
 > $\mathrm{Norm}\left(B,1\right)$
 ${10}$ (4)
 > $h≔⟨3|4⟩$
 ${h}{≔}\left[\begin{array}{cc}{3}& {4}\end{array}\right]$ (5)
 > $\frac{h}{\mathrm{Norm}\left(h,1\right)}$
 $\left[\begin{array}{cc}\frac{{3}}{{7}}& \frac{{4}}{{7}}\end{array}\right]$ (6)
 > $v≔⟨a,b,c⟩$
 ${v}{≔}\left[\begin{array}{c}{a}\\ {b}\\ {c}\end{array}\right]$ (7)
 > $\mathrm{Norm}\left(v,\mathrm{\infty }\right)$
 ${\mathrm{max}}{}\left(\left|{a}\right|{,}\left|{b}\right|{,}\left|{c}\right|\right)$ (8)