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 $\forall M \in \mathcal{M}_{n}(\mathbb{R}), \Phi(M) = A^{\top}M + MA$. a) Show that there exists a unique matrix $B \in \mathcal{M}_{n}(\mathbb{R})$ such that $A^{\top}B + BA = I_{n}$. b) Show that $B$ is symmetric and that $\operatorname{det}(B) > 0$.
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 $\forall M \in \mathcal{M}_{n}(\mathbb{R}), \Phi(M) = A^{\top}M + MA$.
a) Show that there exists a unique matrix $B \in \mathcal{M}_{n}(\mathbb{R})$ such that $A^{\top}B + BA = I_{n}$.
b) Show that $B$ is symmetric and that $\operatorname{det}(B) > 0$.