Define$$H_1=\left( \begin{matrix} -A & B \\ -{{B}^{T}} & 0 \\ \end{matrix} \right),$$
where $A\in {{\mathbb{R}}^{n\times n}}$ is a symmetric positive definite matrix. $B\in {{\mathbb{R}}^{n\times m}}$ is a column full rank matrix with $n<m$. Can we prove that $H_1$ is a Hurwitz Matrix (all the eigenvalues have negative real parts)?
Moreover, let $G\in {{\mathbb{R}}^{n\times n}}$ be symmetric negative definite. Define$$H_2=\left( \begin{matrix} AG & B \\ {{B}^{T}G} & 0 \\ \end{matrix} \right).$$ Can we prove that $H_2$ is a Hurwitz Matrix?