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Revision difference : vector p-norm |
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Version 3 |
| A class of vector norms, called a $p$-norm and denoted $||\cdot||_p$, is defined as |
A class of vector norms, called a $p$-norm and denoted $||\cdot||_p$, is defined as |
| \begin{displaymath} |
\begin{displaymath} |
| ||\,x\,||_p = (|x_1|^p + \cdots + |x_n|^p)^\frac{1}{p}\qquad p\geq1, x\in\mathbb{R}^n |
||\,x\,||_p = (|x_1|^p + \cdots + |x_n|^p)^\frac{1}{p}\qquad p\geq1, x\in\mathbb{R}^n |
| \end{displaymath} |
\end{displaymath} |
| The most widely used are the 1-norm, 2-norm, and $\infty$-norm: |
The most widely used are the 1-norm, 2-norm, and $\infty$-norm: |
| \begin{displaymath} |
\begin{displaymath} |
| \begin{array}{ll} |
\begin{array}{ll} |
| ||\,x\,||_1 & = |x_1| + \cdots + |x_n| \\ |
||\,x\,||_1 & = |x_1| + \cdots + |x_n| \\ |
| ||\,x\,||_2 & = \sqrt{|x_1|^2 + \cdots + |x_n|^2} = \sqrt{x^Tx} \\ |
||\,x\,||_2 & = \sqrt{|x_1|^2 + \cdots + |x_n|^2} = \sqrt{x^Tx} \\ |
| ||\,x\,||_\infty & = \displaystyle\max_{1\leq i\leq n}|x_i| |
||\,x\,||_\infty & = \displaystyle\max_{1\leq i\leq n}|x_i| |
| \end{array} |
\end{array} |
| \end{displaymath} |
\end{displaymath} |
| The 2-norm is sometimes called the Euclidean vector norm, because |
The 2-norm is sometimes called the Euclidean vector norm, because |
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$||\,x-y\,||_2$ yields the Euclidean distance between any two vectors $x,y\in R^n$. The 1-norm is also called the taxicab metric (sometimes Manhattan metric) since the distance of two points can be viewed as the distance a taxi would travel on a city (horizontal and vertical movements).
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$||\,x-y\,||_2$ yields the Euclidean distance between any two vectors $x,y\in R^n$,
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