See also: [[2-norm]] [[Infinity-norm]] # Vector Norms Motivation: Understand how to quantify the "size" of a vector or matrix. ## What is a Vector Norm? A vector norm describes the size, or 'length' of a vector $\vec{x}$. The traditional means of describing a vector norm is actually known as the "[[2-norm]]", which is the Euclidean Norm, but there are actually a lot of different ways of categorizing size of vectors. ## Properties of Vector Norms A vector norm is defined by the following confidtions. 1. A norm $||\vec{x}||$ is always $\geq 0$, with $||\vec{x}|| = 0$ if and only if $||\vec{x}|| = \vec{0}$ 2. $||c\vec{x}|| = |c| \hspace{0.1cm}||\vec{x}||$ for any scalar $c$ 3. The [[Triangle Inequality]] must hold. A p-norm for a vector is, where p is a positive integer defined as: $||\vec{x}||_p = \left[\sum_{n} |\vec{x}|^p \right]^{1/p} $ Some examples: - The 1-norm (Manhattan norm) is the sum of all the elements in a vector. - The [[2-norm]] is the Euclidean length of a vector - The [[Infinity-norm]] is the maximum of absolute values in a vector. [^1] [^1]:https://pages.cs.wisc.edu/~amos/412/lecture-notes/lecture14.pdf