# What is a Norm? Norm of a vector? P-Norm?

• Dec 10th 2009, 01:23 AM
bnr34rb26dett
What is a Norm? Norm of a vector? P-Norm?
so norm of a number gives out the length
norm of a vector gives out the magnitude
what is a p norm and what does it solve for?
• Dec 10th 2009, 02:06 AM
craig
This link may be of help to you

Matrix Norm -- from Wolfram MathWorld
• Dec 10th 2009, 03:22 AM
CaptainBlack
Quote:

Originally Posted by bnr34rb26dett
so norm of a number gives out the length
norm of a vector gives out the magnitude
what is a p norm and what does it solve for?

$\displaystyle \| \bold{x} \|_p = \left( \sum_i |x_i|^p \right)^{1/p}$

CB
• Dec 10th 2009, 06:06 AM
HallsofIvy
In general, a "norm" on a vector space is a function that assigns to each vector a number, ||v||, satifying:
1) ||u+ v||$\displaystyle \le$ ||u||+ ||v||.
2) ||v||= 0 if and only if v is the 0 vector.
3) ||av||= |a|||v|| for any scalar (number) a, and vector v.

The formula Captain Black gives defines the "p" norm, over $\displaystyle R^n$, for any positive number p (p is usually taken to be an integer but that is not necessary)- though I believe it should include an absolute value:
$\displaystyle ||v||= \left(\sum |x_i|^p\right)^{1/p}$.

Of course, the p-norm for p= 2 is the usual "Euclidean norm" $\displaystyle ||v||= \sqrt{\sum x_i^2}$.

If p= 1 we get the "one-norm" $\displaystyle ||v||= \sum |x_i|$.

Sometimes, although it doesn't fit the formula above, we define the "0-norm" to be $\displaystyle ||v||= max |x_i|$- that is, take the absolute value of all components, then select the largest to be the norm.