I have been trying to understand what is the difference between $L_1$ and $L_2$ norm and cant figure it out.
In this webpage I got a clear understanding of why we would use $L_1$ norm (scroll down till you get to the google maps picture).
I went on matlab and calculated the norm for the matrix $A=[3, 7]$ and got that the $L_1$ norm is $10$, which makes sense as in the example above. It is the distance between $(0,0)$ and $(3,7)$. When I do the $L_2$ norm I get $7.61\dots$ and then $L_3$ is $7.1$ and so on until it converges to $7$. What do these calculations mean? Why are the numbers getting smaller and converging to $7$. In my intuition I can grasp why would the distance between $(0,0)$ and $(3,7)$ be $10$, but cant understand the need of $L_2$ and that being $7.61$. I looked everywhere for an intituitive explanation but all I get is how to calculate the norm, which I already know how to do.
Any reason why would it be best to use $L_2$ to calculate the distance-magnitude of a vector and why is the number smaller in $L_2$ than the intituitive case of $L_1$? Thanks.