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# Norms

Modified 2018-06-22 by Andrea Censi

Dzenan Lapandic

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Other metrics can be defined to measure the “length” of a vector. Here, we report some commonly used norms. For a more in depth discussion of what constitutes a norm, and their properties, see:

### $p$-norm

Modified 2018-06-22 by Andrea Censi

Let $p \geq 1 \in \reals$. The $p$-norm is defined as:

$p$-norm \begin{align} \label{eq:vec-p-norm}\tag{1} \|\avec{v}\|_p = \displaystyle \left( \sum_{i=1}^{n} |v_i|^p \right)^{\frac{1}{p}}. \end{align}

The $p$-norm is a generalization of the $2$-norm ($p=2$ in \eqref{eq:p-norm}) introduced above (Definition 5 - Length of a vector). The following $1$-norm and $\infty$-norm can as well be obtained from \eqref{eq:vec-p-norm} with $p=1$ and $p \rightarrow \infty$ respectively.

### One norm

Modified 2018-06-22 by Andrea Censi

The $1$-norm is the sum of the absolute values of a vector’s components. It is sometimes referred to as the Taxicab norm, or Manhattan distance as it well describes the distance a cab has to travel to get from a zero starting point to a final destination $v_i$ on a grid.

$1$-norm Given a vector $\avec{v} \in \reals^n$, the $1$-norm is defined as: \begin{align} \label{eq:vec-one-norm}\tag{2} \|\avec{v}\| = \displaystyle \sum_{i=1}^{n}|v_i|. \end{align}

### $\infty$-norm

Modified 2018-06-22 by Andrea Censi

The infinity norm measures the maximum component, in absolute value, of a vector.

$\infty$-norm \begin{align} \label{eq:vec-inf-norm}\tag{3} \|\avec{v}\| = \displaystyle \max(|v_1|, \cdots, |v_n|). \end{align}

## Definition

Modified 2018-06-22 by Andrea Censi

## Properties

Modified 2018-06-22 by Andrea Censi

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