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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.

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Look at Figure 2 on your web page, just below the Google Earth picture. The $L_1$ norm is the length of the red path. The $L_2$ norm is the length of the blue line segment. –  Rahul Dec 26 '12 at 16:55
One would naturally take the length of a vector to be the $\ell_2$-norm (not the $\ell_1$ norm). This is just the Pythagorean Theorem: take the vector to be the hypotenuse of a right triangle. The other side lengths are the components of your vector. In your case, you have a vector with tail at $(0,0)$ and tip at $(3,7)$. I'ts $\ell_2$-length is $\sqrt{3^2+7^2}$. For other $p$, the $\ell_p$-norms do define a "length" (but not the usual notion of length). –  David Mitra Dec 26 '12 at 17:06

4 Answers 4

up vote 5 down vote accepted

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.

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Great, thanks a lot! this really helps. –  Carlos Dec 26 '12 at 17:51
@Carlos: Glad to hear it; you’re welcome! –  Brian M. Scott Dec 26 '12 at 17:52

Here is the formula for a norm of any number, p. There also exists the Frobenius Norm.

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Thanks a lot for your time and help! So there is no way to grasp what l3, l4, l5 are in an intuitive manner right? –  Carlos Dec 26 '12 at 17:49
@Carlos, I don't know. –  power Dec 27 '12 at 2:44

The $L^2$ norm is a straightforward generalization of the standard Euclidean norm on $\mathbb{R}^n$.

You can ask why $L^2$ norm is so good, or why it is so popular?

The answer is simple: the $L^2$ norm is easiest to play with, it is easiest to perform manipulations with it, and what is perhaps the most important it is easy to calculate. All that good properties of that norm have one simple source: the $L^2$ norm is a norm induced by an inner product on a Hilbert space -$L^2$ is a Hilbert space. And when you are dealing with a Hilbert space you can do almost anything, you have orthonormal bases, orthogonal projections and other Hilbert's space gadgetery. This norm comes with a rich structure that enables you to do many things with ease. That is the true rationale standing behind the popularity of the $L^2$ norm in applications.

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The $L^2$ norm is crow-fly distance and the $L^1$ norm is taxicab distance.

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