Part 18 : Norms
Norm is a function that returns length/size of any vector (except zero vector).
Lets assume a vector x such that
For any function f to be a norm, it has to satisfy three conditions
Condition 1
If norm of x is greater than 0 then x is not equal to 0 (Zero Vector) and if norm is equal to 0 then x is a zero vector.
Condition 2
For any scalar quantity, say K
Condition 3
Assuming that we have another vector y
If these three conditions are satisfied then the function f is a norm.
Commonly used Norms
The most commonly used norms are clubbed under p-norms or (lₚ-norms) family, where p is any number greater than or equal to 1.
The p-norm of vector x will be denoted as
To calculate the p-norm of vector x we have the formula
It could be re-written in simplified form as
Manhattan Distance (1-norm)
1-norm is also called Manhattan distance because it measures distance between two points in a city given that you can only travel along orthogonal city blocks.
Suppose, for a vector a we have to calculate 1-norm.
1-norm could be represented on a figure as
To calculate 1-norm using formula, we could just replace p by 1
Euclidean Norm (2-norm)
The most used norm within p-norm family is the Euclidean Norm or 2-norm. We have used it earlier to calculate the magnitude of vector.
Euclidean Norm returns the shortest distance between two points.
So, 2-norm of vector a will be
and to show 2-norm of vector a on figure
Infinity-norm
The infinity-norm returns maximum absolute value in the given vector.
Infinity-norm of vector a will be
Suppose we have to find infinity-norm of another vector, say b
then
Read Part 19 : Minors and Cofactors
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