what is a magnitude of an element in an vector? how to compute this magnitude?

Question

i am learning the DeepLearningBook, in which chapter 2 says

One other norm that commonly arises in machine learning is the $L^{\infty}$ norm, also known as the max norm. This norm simplifies to the absolute value of the element with the largest magnitude in the vector

$$ \begin{equation} \left\lVert{\boldsymbol{\mathit{x}}}\right\rVert_\infty = \max_i |x_i|. \end{equation} $$

what is a magnitude of an element in an vector? how to compute this magnitude?

Answer 1

The magnitude of an element in a vector is whatever makes sense at the time and will be explicitly defined for whatever context you are working in in the event that it is a more exotic scenario.

Generally, in most normal scenarios where you are working with the $L^\infty$ norm, you would be working in $\Bbb R^n$ or $\Bbb C^n$ or something similar for your vector space in which case the magnitude of a specific entry of a vector is very simply the "absolute value" which we are used to using every day of our lives.

For example if the vector was $\begin{bmatrix}1\\-7\\3\end{bmatrix}$, the $L^\infty$ norm of this vector would be $7$ as this is the largest absolute value of the entries of the vector.

Answer 2

If your vector $x$ is $(x_1,\ldots,x_n)$, then you oick th $i\in\{1,2,\ldots,n\}$ for which $\lvert x_i\rvert$ is the greatest. Then $\lVert x\rVert=\lvert x_i\rvert$.