L1 norm

The L1 norm of a vector is the sum of the absolute values of its coordinates.

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01The L1 norm adds the size of each coordinate#

The L1 norm adds the size of each coordinate. For $v=(-3,2)$ , the horizontal contribution is $3$ and the vertical contribution is $2$ .

For $v=(-3,2)$ , take each absolute value before adding. The negative coordinate contributes $|-3|=3$ .

Add the coordinate magnitudes.

\|v\|_1=3+2=5

The subscript $1$ names this rule. Every entry contributes its absolute value, including entries after the first two.

For a vector with n coordinates.

\|v\|_1=\sum_{i=1}^{n}|v_i|

Check your reasoning

Compute the L1 norm of $v=(-4,1,-2)$ .

Show answer and explanation

The contributions are $4+1+2=7$ .

Do not add signed coordinates first. For $(3,-3)$ , the signed sum is $0$ , but the L1 norm is $3+3=6$ . Only the all-zero vector has L1 norm zero.

A zero entry adds nothing.

\|(0,-5,0)\|_1=5

Check your reasoning

Three controls change by −2, 5 and −1. What is the total unsigned adjustment?

Show answer and explanation

Add the magnitudes of the three changes: $2+5+1=8$ .

Check your reasoning

For $v=(-2,5)$ , repair the L1 rule $|-2+5|$ .

Show answer and explanation

Take each absolute value.

Add $2+5=7$ .

Key takeaway