norm - Script command

Returns the natural norm of a matrix induced by the L2-norm (spectral norm). For a matrix A this is the square root of the maximum eigenvalue of the matrix product A ^H A, where A ^H is the conjugate transpose of A.

Note that for a N-dimensional complex vector x = [x ₁ ,x ₂ ,...,x _N ] this reduces to the usual norm:

$$ \|x\|=\sqrt{\sum_{i=1}^{N} x_{i}^{*} x_{i}}=\sqrt{\sum_{i=1}^{N}\left|x_{i}\right|^{2}} $$

Example

Find the usual norm of real and complex vectors.

y1=[1,2,3];
y2=[1+1i,2,3]; #y2 has complex elements
?norm(y1);
?norm(y2);
result: 
3.74166  
result: 
3.87298  
# Confirm the results with the usual definition:
?sqrt(sum(conj(y1)*y1));
?sqrt(sum(conj(y2)*y2));
result: 
3.74166  
result: 
3.87298+0i  

Find the usual norm of a complex matrix.

A=[1,2+7i,3;7+3i,0,9];
?norm(A);
?sqrt(max(eig(mult(ctranspose(A),A)))); # confirm the result using the definition
result: 
12.332  
result: 
12.332  

See Also

List of commands , sqrt , sum , conj , max , eig , mult , ctranspose