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 $x(A_1+A_2)x^T = xA_1x^T + xA_2x^T > 0$.