Manhattan Norm (L₁) Calculator

About Manhattan Norm (L₁)

The Manhattan norm, also known as L₁ norm or Taxicab norm, measures the sum of the absolute values of a vector's components. It's named after the Manhattan distance, which represents the distance a taxi would drive in a city laid out in a grid-like pattern.

Formula:

For a vector \(\vector{x} = (x_1, x_2, \ldots, x_n)\):
\[ \norm{\vector{x}}_1 = \sum_{i=1}^n \abs{x_i} = \abs{x_1} + \abs{x_2} + \cdots + \abs{x_n} \]

Properties:

Non-negativity: \(\norm{\vector{x}}_1 \geq 0\)
Positive definiteness: \(\norm{\vector{x}}_1 = 0\) if and only if \(\vector{x} = \vector{0}\)
Homogeneity: \(\norm{\alpha\vector{x}}_1 = \abs{\alpha}\norm{\vector{x}}_1\)
Triangle inequality: \(\norm{\vector{x} + \vector{y}}_1 \leq \norm{\vector{x}}_1 + \norm{\vector{y}}_1\)

Applications:

Urban distance calculation (city block distance)
Error estimation in statistics
Feature selection in machine learning
Sparse approximation problems
Compressed sensing

Example:

For the vector \(\vector{x} = (3, -4, 2)\):
\[ \norm{\vector{x}}_1 = \abs{3} + \abs{-4} + \abs{2} = 3 + 4 + 2 = 9 \]

Manhattan Norm Calculator

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About Manhattan Norm (L₁)

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