What is the reasoning behind: $\sup_{\|y\|_{2} \leq 1} \theta \mathbf{x}^T \mathbf{y} = |\theta| \sup_{\|y\|_{2}\leq 1} {\mathbf{x}^T \mathbf{y}}$

Question

What is the reasoning behind this equality:

$$\sup_{\|y\|_{2} \leq 1} \theta \mathbf{x}^T \mathbf{y} = |\theta| \sup_{\|y\|_{2}\leq 1} {\mathbf{x}^T \mathbf{y}}$$

where $\mathbf{x} $ and $\mathbf{x} \in \mathbb{R}^n$.

Thanks for your help.

Answer 1

Note that $\lVert y \rVert = \lVert -y \rVert$. This implies the maximizer is non-negative (why? if it were negative, just replace $y$ with $-y$ and get something positive). If $\theta$ is negative, we can write $\theta = - |\theta|$ and if $y$ is the maximizer, replace $y$ with $-y$ and $\theta$ with $-\theta$ and get the same result.