Find a lower bound on $\min_{\boldsymbol{x}}(\boldsymbol{x}^\textrm{T}\boldsymbol{A}\boldsymbol{y})^2$

Question

I wonder if there is a good way to find the lower bound of the following term: \begin{equation} \min_{\boldsymbol{x}}\,(\boldsymbol{x}^\textrm{T}\boldsymbol{A}\boldsymbol{y})^2 \end{equation} where $\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^n$ have unit norm, and $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ is a symmetric matrix. $\boldsymbol{A}$ and $\boldsymbol{y}$ are known. Any help would be greatly appreciated! Thanks!

******edit****** Thanks guys! It is zero.

Answer 1

The minimum is $0$, because it's possible to pick $x$ to be orthogonal to $Ay$.