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  is .

The infinity-norm of  Vector V is .

Maple implements Vector norms for all .  For  the final pth root computation is not done, that is, the calculation is . This defines a metric on , but the pth root is not a norm and the form computed by Norm in such cases is more useful.  The limiting case of  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  and the special case  (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.