This article presents an $H$-matrix approach to investigate the distributed control design problems for a higher order quasistrongly connected signed network with linear dynamics and heterogeneous control gains. By tuning the control gains based only on the neighboring information of agents, an adaptive algorithm is proposed to ensure that the network can achieve the fully distributed control objective without employing any global information of the network. When the rooted subgraph is structurally unbalanced, it is proved that the Laplacian matrix is a nonsingular $H$-matrix, thus making the network stable. When the rooted subgraph is structurally balanced, some properties of constrained Rayleigh quotients and Lyapunov diagonally stable $H$-matrices are utilized to show that the network can reach bipartite (or interval bipartite) consensus once the network is structurally balanced (or unbalanced). Furthermore, the results of the adaptive algorithm are applied to develop distributed control design for signed networks with heterogeneous constant control gains.