  This paper deals with linear algebraic equations where the global coefficient
matrix and constant vector are given respectively, by the summation of the
coefficient matrices and constant vectors of the individual agents. Our
approach is based on reformulating the original problem as an unconstrained
optimization. Based on this exact reformulation, we first provide a
gradient-based, centralized algorithm which serves as a reference for the
ensuing design of distributed algorithms. We propose two sets of exponentially
stable continuous-time distributed algorithms that do not require the
individual agent matrices to be invertible, and are based on estimating
non-distributed terms in the centralized algorithm using dynamic average
consensus. The first algorithm works for time-varying weight-balanced directed
networks, and the second algorithm works for general directed networks for
which the communication graphs might not be balanced. Numerical simulations
illustrate our results.
