  This paper investigates the distributed computation of the well-known linear
matrix equation in the form of $AXB = F$, with the matrices A, B, X, and F of
appropriate dimensions, over multi-agent networks from an optimization
perspective. In this paper, we consider the standard distributed
matrix-information structures, where each agent of the considered multi-agent
network has access to one of the sub-block matrices of A, B, and F. To be
specific, we first propose different decomposition methods to reformulate the
matrix equations in standard structures as distributed constrained optimization
problems by introducing substitutional variables; we show that the solutions of
the reformulated distributed optimization problems are equivalent to least
squares solutions to original matrix equations; and we design distributed
continuous-time algorithms for the constrained optimization problems, even by
using augmented matrices and a derivative feedback technique. Moreover, we
prove the exponential convergence of the algorithms to a least squares solution
to the matrix equation for any initial condition.
