compute the p-norm of a Matrix or Vector
Norm(A, p, options)
Matrix or Vector
(optional) non-negative number, infinity, Euclidean, or Frobenius; norm selector that is dependent upon A
(optional) parameters; for a complete list, see LinearAlgebra[Norm]
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.
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<∞ is add⁡Vip,i=1..Dimension⁡V1p.
The infinity-norm of Vector V is max⁡seq⁡Vi,i=1..Dimension⁡V.
Maple implements Vector norms for all 0≤p≤∞. For 0<p<1 the final pth root computation is not done, that is, the calculation is add⁡Vip,i=1..Dimension⁡V. 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=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.
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=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.
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