Distributed algorithms of solving linear matrix equations via double-layered networks

This paper focuses on designing distributed algorithms for solving the linear matrix equation in the form of AXB=F via a double-layered multi-agent network. Firstly, we equivalently transform the least squares (LS) problem of AXB=F into two subproblems. We introduce a double-layered multi-agent network, in which each agent only has access to partial information of matrices A, B and F, to solve these two subproblems in a distributed structure. Subsequently, we develop continuous-time and discrete-time distributed algorithms to obtain an LS solution of AXB=F. In particular, the sufficient and necessary conditions about the step-sizes are established, under which the discrete-time algorithm can linearly converge to an LS solution. Compared with the previous results in which each agent needs to communicate multiple matrix variables with the neighbors, the proposed algorithms effectively reduce communication burden for each agent since only one matrix variable is communicated in the upper-layer network and the communication variables in the lower-layered network are vectors with lower dimensions than matrices. Finally, an application example is provided to demonstrate the effectiveness of the proposed algorithms.