p-Norm and Unit Circle Pics

This was just out of curiosity and now it seems like a nice way to prove that any vector’s unit circle $\infty$-norm is a square, which is nothing but the maximum of all the scalars in that vector.

Well, this is mainly because am undergoing a course on Matrix Algebra right now, and wanted to experiment using Matlab somehow! 🙂 So, a norm or more generally the p-norm for a vector is basically

$\|x\|_p = (|x_1|^p + |x_2|^p + ... + |x_N|^p)^{1/p}$

Yes, and am bloody glad that I can use $\LaTeX$ code in WordPress 🙂 So nice of them. Here’s how its done. Just insert the lines $latex \|x\|_p = (|x_1|^p + |x_2|^p + ... + |x_N|^p)^{1/p}$ in your required section. No space is to be put between \$ and latex, I couldn’t show without it though as it was converting to $\LaTeX$.

Anyways, so to find the unit circle (only 1 quadrant) I just did a scatter plot of the random number vectors whose norm was less than 1, and it looks like this.

Unit Circles of Norm - Matlab Simulations

This will let you know that the unit-circle went from a triangle like shape, to circle and then is bulging towards the (1,1) point. It reaches this only at $\infty$. Note that this is just in the positive quadrant, but its symmetrical in all quadrants of the 2-D space considered here.

10 thoughts on “p-Norm and Unit Circle Pics”

1. Vanamali

Just to add : The unit circle of p-norm, is the set of all vectors with their p-norm =1 ( in this case, the vectors refer to points in R^2 )

2. Chacha

Can any one help me the matlab codes for plotting p-norms in three dimension for norm(×)<=1
i.e
m=3
p=1-5

3. chacha

This site is very useful but i would like to know the matlab codes for geneting p_norms.
for m=3 and p=12,3,4,5 and infinity
Given norm(×)<=1