Why we use the 2-norm all over the place instead of the 1-norm although both are equivalent and the 1-norm is more effieciently computable? I am currently implementing an algorithm that normalizes a vector in each step (using 2-norm). So i could also just use the 1-norm instead? More details: power iteration for the calculation of the dominat eigenvalue of a matrix:
x:=random vector of length numrows(A) x=x/||x||
while l changes (e.g. abs(l-l_last)<tolerance)
x=Av
l=||x||
x=x/||x||
On
In your use case, the actual norm of the vector does not matter since you are only concerned about the dominant eigenvalue. The only reason to normalize during the iteration is to keep the numbers from growing exponentially. You scale the vector however you want to prevent numeric overflow.
A key concept about eigenvectors and eigenvalues is that the set of vectors corresponding to an eigenvalue form a linear subspace. This is a consequence of multiplication by a matrix being a linear map. In particular, any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.
The Wikipedia article Power method mentions the use of the Rayleigh quotient to compute an approximation to the dominant eigenvalue. For real vectors and matrices it is given by the value $\, (v\cdot Av)/(v \cdot v). \,$ There are probably good reasons for the use of this formula. Of course, if $\,v\,$ is normalized so that $\, v \cdot v = 1, \,$ then you can simplify that to $\, v\cdot Av. \,$
On
Tell me which of these looks more like a ball, and I'll tell you which one is the better norm:
$$ \ $$
More seriously, the two-norm is given by an inner product, and this has far reaching consequences: like orthogonality, projections, complemented subspaces, orthonormal bases, etc., etc., etc., features that we see in Hilbert spaces.
In the eigenvalue algorithms, you're attempting to generate the eigenvalues and the orthogonal eigenvectors. The step right there is the normalization part. If you look at this.
the next step doesn't work with the $\ell_{1}$ norm. Or the actual purpose of it doesn't. The entire purpose here was to make that normalized. If you realize how these algorithms work if the eigenvalues are closer together it makes them take a lot longer. So numeric error will destroy the algorithm.