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 $A \in \mathcal{M}_{n}(\mathbb{R})$ such that $A_{s}$ is positive definite. a) Let $X = Y + \mathrm{i}Z$ be a column matrix of $\mathcal{M}_{n,1}(\mathbb{C})$, where $Y$ and $Z$ belong to $\mathcal{M}_{n,1}(\mathbb{R})$. We set $\bar{X} = Y - \mathrm{i}Z$ and we identify the matrix $\bar{X}^{\top}AX \in \mathcal{M}_{1}(\mathbb{C})$ with the complex number equal to its unique entry. Show that, if $X \neq 0$, then $\operatorname{Re}(\bar{X}^{\top}AX) > 0$, where $\operatorname{Re}(z)$ denotes the real part of $z \in \mathbb{C}$. b) Show that $A$ 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 $A \in \mathcal{M}_{n}(\mathbb{R})$ such that $A_{s}$ is positive definite.
a) Let $X = Y + \mathrm{i}Z$ be a column matrix of $\mathcal{M}_{n,1}(\mathbb{C})$, where $Y$ and $Z$ belong to $\mathcal{M}_{n,1}(\mathbb{R})$. We set $\bar{X} = Y - \mathrm{i}Z$ and we identify the matrix $\bar{X}^{\top}AX \in \mathcal{M}_{1}(\mathbb{C})$ with the complex number equal to its unique entry.
Show that, if $X \neq 0$, then $\operatorname{Re}(\bar{X}^{\top}AX) > 0$, where $\operatorname{Re}(z)$ denotes the real part of $z \in \mathbb{C}$.
b) Show that $A$ is positively stable.