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 ${\mathrm{add}\left({\left|V_i\right|}^p\&comma;i=1..\mathrm{Dimension}\left(V\right)\right)}^{\frac{1}{p}}$.

The infinity-norm of  Vector V is $\max \left(\mathrm{seq}\left(\left|V_i\right|\&comma;i=1..\mathrm{Dimension}\left(V\right)\right)\right)$.

Maple implements Vector norms for all $0\le p\le \mathrm{\infty }$.  For $0<p<1$ the final pth root computation is not done, that is, the calculation is $\mathrm{add}\left({\left|V_i\right|}^p\&comma;i=1..\mathrm{Dimension}\left(V\right)\right)$. 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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\&colon;$

$A≔⟨⟨1\&comma;-1\&comma;0⟩\left|⟨0\&comma;1\&comma;1⟩\right|⟨1\&comma;0\&comma;1⟩⟩$

$A≔\left[\begin{array}{ccc}1& 0& 1\\ \mathrm{-1}& 1& 0\\ 0& 1& 1\end{array}\right]$

$\mathrm{Norm}\left(A\&comma;2\right)$

$\sqrt{3}$

$B≔⟨⟨10\&comma;0\&comma;0⟩\left|⟨0\&comma;9\&comma;\frac{1}{2}⟩\right|⟨2\&comma;4\&comma;1⟩⟩$

$B≔\left[\begin{array}{ccc}10& 0& 2\\ 0& 9& 4\\ 0& \frac{1}{2}& 1\end{array}\right]$

$\mathrm{Norm}\left(B\&comma;1\right)$

$10$

$h≔⟨3|4⟩$

$h≔\left[\begin{array}{cc}3& 4\end{array}\right]$

$\frac{h}{\mathrm{Norm}\left(h\&comma;1\right)}$

$\left[\begin{array}{cc}\frac{3}{7}& \frac{4}{7}\end{array}\right]$

$v≔⟨a\&comma;b\&comma;c⟩$

$v≔\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$

$\mathrm{Norm}\left(v\&comma;\mathrm{\infty }\right)$

$\max \left(\left|a\right|\&comma;\left|b\right|\&comma;\left|c\right|\right)$