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.
Best Answer
The $1$-norm and $2$-norm are both quite intuitive. The $2$-norm is the usual notion of straight-line distance, or distance ‘as the crow flies’: it’s the length of a straight line segment joining the two points. The $1$-norm gives the distance if you can move only parallel to the axes, as if you were going from one intersection to another in a city whose streets run either north-south or east-west. For this reason it’s sometimes called the taxicab norm, and the associated distance the taxicab distance.
The $n$-norms for $n>2$ don’t correspond to anything very intuitive. However, as $n$ increases they do approach the $\infty$-norm, which is simply the maximum of the absolute values of the coordinates. The $\infty$-norm of $(3,7)$, for instance, is the maximum of $|3|$ and $|7|$, which of course is $7$. To find the $\infty$-norm distance between two points in the plane, see how far apart they are in the east-west direction (parallel to the $x$-axis) and how far apart they are in the north-south direction (parallel to the $y$-axis), and take the larger of those numbers. This is a bit like the taxicab distance: it’s as if you only had to pay for your east-west distance or your north-south distance, whichever was larger.