In my coursework for Linear Algebra this semester, we were introduced to norm, which was essentially the length of a vector.
For a vector $x = [x_1, x_2, \dots, x_n]$, the norm is given by:-
$$ ||x|| = \sqrt[2]{|x_1|^2 + |x_2|^2 + \dots + |x_2|^p} $$
The issue was, in my quest to solve solitaire with my friend, I barely listened to the lecture. Off to self learn.
And when I did look up, I found out that there are more than 1 "norms". Infact, there are infinitely many, and these come under the Minkowski Norms, or $L_p$ norm.
Minkowski norms can be generalised into a formula, and each norm can be generated from it by changing the parameter $p$.
$$ ||x||_p = \sqrt[p]{|x_1|^p + |x_2|^p + \dots + |x_n|^p} $$
The parameter $p$ can be any real number $p \ge 1$.
And the reason it's greater than 1, is to satisfy the triangle inequality. If you recollect, it states that "The sum of two sides of a triangle is greater than the third side".
Mathematically, this is represented as:- $$ ||u + v|| \le ||u|| + ||v|| $$
There is a generalised form of this as well, called the "Minkowski Triangle Inequality", to expand it into $L_p$ spaces, which is:-
$$ (\sum_{i=1}^{n} |x_i + y_i|^p)^{\frac{1}{p}} \le (\sum_{i=1}^{n} |x_i|^p)^{\frac{1}{p}} + (\sum_{i=1}^{n} |y_i|^p)^{\frac{1}{p}} $$
While the standard triangle inequality is limited to Euclidean space ($L_2$ norm), the Minkowski Triangle Inequality is applicable to all $L_p$ spaces.
The most common Minkowski Norms are:-
- $L_1$ norm (Manhattan norm) - Sum of absolute differences $||x||_1 = \sum_{i=1}^{n} |x_i|$
- $L_2$ norm (Euclidean norm) - Standard straight line difference $||x||_2 = \sqrt{\sum_{i=1}^{n} x_i^2}$
- $L_\infty$ norm (Chebyshev norm) - Maximum absolute difference $||x||_\infty = \max_{i=1}^{n} |x_i|$
While we haven't explored $L_p$ spaces in depth, they are essentially vector spaces of functions equipped with the $L_p$ norm. More formally, an $L_p$ space consists of all functions whose absolute value raised to the $p$-th power is integrable.