Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix

$$ M = \begin{bmatrix} A & B & 0\\ B^{\top} & C & -B^{\top} \\ - A & -B & A\end{bmatrix} $$ has eigenvalues with positive real part. Numerical tests suggest this is true, but I cannot prove it.

Edit It is not true that $M + M^\top$ has positive eigenvalues, i.e. that $\langle x, Mx\rangle \geq 0$ for all $x$, which is (?) a sufficient condition for $M$ to have eigenvalues with positive real parts.

Edit 2 After further numerical tests, letting $\lambda_{\min}$ the smallest eigenvalue of $N$, and $v_{\min}$ the smallest real part of the eigenvalues of $M$, it seems like we should have a bound like $$ v_{\min} \geq \rho \cdot \lambda_{\min} $$ where the constant $\rho$ is $\simeq 0.963$.

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix

$$ M = \begin{bmatrix} A & B & 0\\ B^{\top} & C & -B^{\top} \\ - A & -B & A\end{bmatrix} $$ has eigenvalues with positive real part. Numerical tests suggest this is true, but I cannot prove it.

Edit It is not true that $M + M^\top$ has positive eigenvalues, i.e. that $\langle x, Mx\rangle \geq 0$ for all $x$, which is (?) a sufficient condition for $M$ to have eigenvalues with positive real parts.

Edit 2 After further numerical tests, letting $\lambda_{\min}$ the smallest eigenvalue of $N$, and $v_{\min}$ the smallest real part of the eigenvalues of $M$, it seems like we should have a bound like $$ v_{\min} \geq \rho \cdot \lambda_{\min} $$ where the constant $\rho$ is $\simeq 0.963$.

Edit 3 I expect that we can answer the question by finding the right factorization for $M$. For instance it is easy to show that $$ \begin{bmatrix} A & B & -A\\ B^{\top} & C & -B^{\top} \\ - A & -B & A\end{bmatrix} $$ has positive eigenvalues since it equals $ \begin{bmatrix} I & 0 & -I \\ 0& I & 0\end{bmatrix}^{\top}N \begin{bmatrix} I & 0 & -I \\ 0& I & 0\end{bmatrix}$

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $\begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix

$$ M = \begin{bmatrix} A & B & 0\\ B^{\top} & C & -B^{\top} \\ - A & -B & A\end{bmatrix} $$ has eigenvalues with positive real part. Numerical tests suggest this is true, but I cannot prove it.

Edit It is not true that $M + M^\top$ has positive eigenvalues, i.e. that $\langle x, Mx\rangle \geq 0$ for all $x$, which is (?) a sufficient condition for $M$ to have eigenvalues with positive real parts.

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix

$$ M = \begin{bmatrix} A & B & 0\\ B^{\top} & C & -B^{\top} \\ - A & -B & A\end{bmatrix} $$ has eigenvalues with positive real part. Numerical tests suggest this is true, but I cannot prove it.

Edit It is not true that $M + M^\top$ has positive eigenvalues, i.e. that $\langle x, Mx\rangle \geq 0$ for all $x$, which is (?) a sufficient condition for $M$ to have eigenvalues with positive real parts.

Edit 2 After further numerical tests, letting $\lambda_{\min}$ the smallest eigenvalue of $N$, and $v_{\min}$ the smallest real part of the eigenvalues of $M$, it seems like we should have a bound like $$ v_{\min} \geq \rho \cdot \lambda_{\min} $$ where the constant $\rho$ is $\simeq 0.963$.