1-norm
High school
Consider a vector $x = (x_1, \dots, x_n)$. The 1-norm of $x$ is the sum of the absolute values of its entries:
\[\norm{x}_1 = \abs{x_1} + \dots + \abs{x_n}.\]Examples
Worked examples
- The vector $x = (1, 1) \in \reals^2$ : $\norm{x}_1 = \abs{1} + \abs{1} = 2$ High school
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The vector $x = (3, -4) \in \reals^2$
: $\norm{x}_1 = \abs{3} + \abs{-4} = 7$
High school
The 1-norm of $(3, -4)$ is $7$, the distance walking $3$ units sideways and then $4$ units down, and its infinity norm, $4$, is the larger of the two moves.
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The vector $x = (5, 0) \in \reals^2$
: $\norm{x}_1 = \abs{5} + \abs{0} = 5$
High school
All three norms of $(5, 0)$ equal $5$, because the vector lies along an axis: the norms differ only for vectors that point between the axes.
Properties
- In the plane, $\norm{x - y}1$ is the distance from $y$ to $x$ when traveling only along horizontal and vertical streets, as in a city grid. The is why the 1-norm is sometimes called the _taxicab norm or Manhattan norm.
- For every $x \in \reals^n$, $\norm{x}_{\infty} \leq \norm{x}_2 \leq \norm{x}_1$.
Distinctions involving this term
- Types of norms: A norm measures the size of a vector, and $\reals^n$ has several common norms that measure size in different ways. The Euclidean norm is the straight-line length, the 1-norm adds the sizes of the entries, and the infinity norm takes the largest entry. Each is a norm, so each satisfies the triangle inequality, and for every vector, $\norm{x}_{\infty} \leq \norm{x}_2 \leq \norm{x}_1$.