A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. We consider the endomorphism $\Phi$ of $\mathcal{M}_{n}(\mathbb{R})$ such that $$\forall M \in \mathcal{M}_{n}(\mathbb{R}), \quad \Phi(M) = A^{\top}M + MA$$ Show that $\Phi$ is positively stable, that is, its matrix in any basis of $\mathcal{M}_{n}(\mathbb{R})$ is positively stable.
A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. We consider the endomorphism $\Phi$ of $\mathcal{M}_{n}(\mathbb{R})$ such that
$$\forall M \in \mathcal{M}_{n}(\mathbb{R}), \quad \Phi(M) = A^{\top}M + MA$$
Show that $\Phi$ is positively stable, that is, its matrix in any basis of $\mathcal{M}_{n}(\mathbb{R})$ is positively stable.