What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$?

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What is the difference between the notations $\|x\|_1, \|x\|_2$ and $|x|$? I think $|x|$ is the magnitude of $x$.

Stephen Philip

Posted 2020-01-26T08:13:10.457

Reputation: 175

Answers

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$\|x\| = |x|$ denotes the absolute value norm, which is a special case of the $L_1$ norm defined on the 1-D vector spaces formed by real or complex numbers.

$\|\textbf{x}\|_1 = \sum_{i=1}^n|x_i|$ denotes the Taxicab / Manhattan norm, relating to how a Taxi would drive along a rectangular grid of roads to reach a point $(x, y)$ from $(0,0)$.

$\|\textbf{x}\|_2 = \sqrt{x_1^2 + \dots + x_n^2}$ denotes the Euclidean norm on an N-D Euclidean space, which is a result of the Pythagorean theorem (the shortest distance between two points).

SoumyadeepB

Posted 2020-01-26T08:13:10.457

Reputation: 335

Could you please elucidate on absolute value norm computation. – Stephen Philip – 2020-01-27T05:18:41.517

1The Absolute value norm is a special case of the $L_1$ norm (defined in 1D). So, computing it would be equivalent to evaluating the absolute value of the single component of the vector. – SoumyadeepB – 2020-01-27T10:54:27.220