Is a Hurwitz matrix minus a positive-definite matrix still a Hurwitz matrix? In a certain multi-agent systems, the $i$th system can be described by the form
\begin{equation}
\begin{array}{cc}
   \left\{ {\begin{array}{l}
    {{\dot x}_i}=A_i{x_i} +B_iu_i\\
    y_i = C_ix_i
    \end{array}} \right.,& i=1,2,\cdots,n
   \end{array}
\end{equation}
The information of two neighbours is
        \begin{equation}
   z_i=-\sum\limits_{j\in n_i}l_{ij}y_j
  \end{equation}
        The output feedback controller is
        \begin{equation}
   \left\{
   \begin{array}{l}
   \dot{\eta}_i=M_i\eta_i+N_iz_i\\
   u_i=O_i\eta_i
   \end{array}
   \right.
  \end{equation}
    The $i$th agent's augmented system is
        \begin{equation}
   \left\{\begin{array}{l}
   \left[\begin{array}{c}
   \dot{x}_i\\
   \dot{\eta}_i
   \end{array}\right]=\left[\begin{array}{cc}
   A_i & B_iO_i\\
   0 & M_i
   \end{array}\right]\left[\begin{array}{c}
   x_i\\
   \eta_i
   \end{array}\right]+\left[\begin{array}{c}
   0\\
   N_i
   \end{array}\right]z_i\\
   y_i=\left[\begin{array}{cc}
   C_i & 0
   \end{array}\right]\left[\begin{array}{c}
   x_i\\
   \eta_i
   \end{array}\right]
   \end{array}\right.
  \end{equation}
        Suppose that
        \begin{equation}
   \bar{x}_i=\left[\begin{array}{c}
   x_i\\
   \eta_i
   \end{array}\right], \bar{A}_i=\left[\begin{array}{cc}
   A_i & B_iO_i\\
   0 & M_i
   \end{array}\right], \bar{B}_i=\left[\begin{array}{c}
   0\\N_i
   \end{array}\right], \bar{C}_i=\left[\begin{array}{cc}
   C_i & 0
   \end{array}\right]
  \end{equation}
Therefore, the whole system of $n$ agents is
        \begin{equation}
  \begin{array}{rl}
   \dot{\tilde{x}}&=\left[\left[\begin{array}{ccc}
   \bar{A}_1 & \cdots & 0\\
   \vdots & \ddots & \vdots\\
   0 & \cdots & \bar{A}_n
   \end{array}\right]-\left[\begin{array}{ccc}
   l_{11}\bar{B}_1\bar{C}_1 & \cdots & l_{1n}\bar{B}_1\bar{C_n}\\
   \vdots & \ddots & \vdots\\
   l_{n1}\bar{B}_n\bar{C}_1 & \cdots & l_{nn}\bar{B}_n\bar{C}_n
   \end{array}\right]\right]\tilde{x}\\
   & \\
   &=\tilde{A}\tilde{x}
   \end{array}
  \end{equation}
where $A_i$ and $M_i$ are Hurwitz, and $\bar{B}_i=\bar{C}_i^T$.
So, $\left[\begin{array}{ccc}
   \bar{A}_1 & \cdots & 0\\
   \vdots & \ddots & \vdots\\
   0 & \cdots & \bar{A}_n
   \end{array}\right]$ is Hurwitz, and $\left[\begin{array}{ccc}
   l_{11}\bar{B}_1\bar{C}_1 & \cdots & l_{1n}\bar{B}_1\bar{C_n}\\
   \vdots & \ddots & \vdots\\
   l_{n1}\bar{B}_n\bar{C}_1 & \cdots & l_{nn}\bar{B}_n\bar{C}_n
   \end{array}\right]$ is positive-definite.
Is $\tilde{A}$ a Hurwitz matrix?
Thank you!

If you are asking whether the sum of two positive definite matrices is still positive definite, then the answer is yes. As when $xA_1x^T>0$ and $xA_2x^T>0$, one have $x(A_1+A_2)x^T = xA_1x^T + xA_2x^T > 0$.
