# In-class quiz 2


CS 59000-NMC, 30 August 2011

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

When is the inequality an equality?

## Solution

The first two are equal when $\max_i |x_i| = \sum_i |x_i|.$ This means that $\vx$ can have only a single non-zero component, otherwise the sum will always be greater.

The second two are equal when $\sum_i |x_i| = n |x_{\max}|$. Put another way, this means that the average magnitude must be equal to the maximum magnitude. This will only happen when $|x_{\max}| = |x_i|$ for all $i$. So the vector must have elements with equal magnitude, but possibly different signs. Over $\CC$, we can change any element by a complex rotation $e^{\imath \theta}$, which does not alter the magnitude.