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Consider matrices $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, such that $A$ has at least $1$ strictly positive eigenvalue.

Let $X_1, X_2 \in \mathbb{R}^{n \times n}$ such that $X_1, X_2 \succ 0$ (positive definite) and, respectively for each $i \in \{1,2\}$, $\exists Y_i \in \mathbb{R}^{m \times n} $ such that:

$$ (A+B Y_i)^\top X_i + X_i (A+BY_i) \preccurlyeq 0 \ \text{ (negative semidefinite)} $$

(1) Find a case in which $\nexists Y \in \mathbb{R}^{m \times n}$ such that we simultaneously have

$$ (A+B Y)^\top X_1 + X_1 (A+B Y) \preccurlyeq 0 \ \text{ and } \ (A + B Y)^\top X_2 + X_2 (A+BY) \preccurlyeq 0 $$

(2) Find $x \in \mathbb{R}^n$ such that: $\nexists y \in \mathbb{R}^m$ such that we simultaneously have

$$ x^\top X_1 (Ax+B y) \leq 0 \ \text{ and } \ x^\top X_2 (Ax+B y) \leq 0 $$

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