Unit-Norm Vectors under Different p-Norms

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows how unit-norm vectors look under different -norms, which are standard norms for finite-dimensional spaces.

[more]

In mathematics, a norm is a function that assigns a length (or size) to a vector. The vector is an object in a vector space, and can thus be a function, matrix, sequence, and so on. A -norm is a norm on a finite-dimensional space of dimension defined as

.

This Demonstration shows sets of unit-norm vectors for different -norms.

The norm for is called the Manhattan or taxicab norm because represents the driving distance from the origin to following a rectangular street grid

The norm for is the usual Euclidean square norm obtained using the Pythagorean theorem

The norm for is simply the maximum over and ,

Vectors ending on the red lines are of unit norm in the corresponding -norm.

[less]

Contributed by: Jelena Kovacevic (June 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details

References

[1] M. Vetterli, J. Kovačević, and V. K. Goyal, Foundations of Signal Processing, Cambridge: Cambridge University Press, 2014. www.fourierandwavelets.org.

[2] Wikipedia. "Norm." (Jun 12, 2012) en.wikipedia.org/wiki/Norm_%28 mathematics %29.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send