suppose $x=\Delta$, $y=M \Phi \Delta$, where $\Delta\in N\times 1$, $M^T=M \in N \times N$ and $\Phi^T=\Phi \in N \times N$. Define $Z=xy^T+yx^T$. It is known from the answer to my previous question that $Z$ has two eigenvalues, one is positive and the other is negative and they are given by

$y^Tx+\sqrt{x^Txy^Ty}$ and $y^Tx-\sqrt{x^Txy^Ty}$.

My question is if we have the following result:

$y^Tx+\sqrt{x^Txy^Ty}=\mathcal{O}(\|\Delta\|^2)$, as $\|\Delta\|\rightarrow \infty$.

Based on my simulations, if increases the magnitude of the elements of $\Delta$, $y^Tx+\sqrt{x^Txy^Ty}$ grows linearly w.r.t. the increases of $\|\Delta\|^2$.