# In-class quiz 2

CS 59000-NMC, 30 August 2011

Your name:

Please answer the following questions. You may not use any outside references or technology. Justify and explain all answers. This quiz is for my own evaluation, so that I can provide better instruction in the course.

## Question

Consider the vector 1-norm. Show that

`\normof{\vec{x}}_{\infty} \le \onormof{\vec{x}}{1} \le n \| \vec{x} \|_{\infty}.`

When is the inequality an equality?

## Solution

`\normof{\vec{x}}_{\infty} = \max_i |x_i| \le \sum_i |x_i| = \normof{\vec{x}}_{1} \le \sum_i |x_{\max}| \le n \normof{\vec{x}}_\infty}`

The first two are equal when This means that can have only a single non-zero component, otherwise the sum will always be greater.

The second two are equal when . Put another way, this means that the average magnitude must be equal to the maximum magnitude. This will only happen when for all . So the vector must have elements with equal magnitude, but possibly different signs. Over , we can change any element by a complex rotation , which does not alter the magnitude.