Let
$$ A=\begin{bmatrix} P & \alpha x\\ -y^\top & 0\end{bmatrix}$$
where $P \in \mathbb{R}^{n \times n}$ is Hurwitz (the eigenvalues of $P$ have strictly negative real parts), $x, y \in \mathbb{R}^{n}$, and $\alpha$ is a real positive scalar. Find the condition for $A$ to be Hurwitz for sufficiently small $\alpha > 0$.
I have analyzed that this is true whenever $y^\top P^{-1} x < 0$ holds. However, I am not able to prove this fact. Help in this regard would be appreciated.
Since $P$ is Hurwitz it is invertible. Hence
$$ A \textrm{ is Hurwitz.} \iff \underbrace{\begin{bmatrix} I& 0 \\y^TP^{-1} & 1\end{bmatrix}}_{\det(T)\textrm{ is 1}} A = TA = \begin{bmatrix} P & \alpha x\\ 0 & \alpha y^TP^{-1}x \end{bmatrix} \textrm{ is Hurwitz.} $$ since $\alpha>0$ this leads to the condition you have given.