Network Flows That Solve Sylvester Matrix Equations

In this article, we study methods to solve a Sylvester equation in the form of
$\mathbf {A}\mathbf {X}+\mathbf {X}\mathbf {B}=\mathbf {C}$
for given matrices
$\mathbf {A}, \mathbf {B}, \mathbf {C}\in \mathbb {R}^{n\times n}$
, inspired by the distributed linear equation flows. The entries of
$\mathbf {A}, \mathbf {B},$
and
$\mathbf {C}$
are separately partitioned into a number of pieces (or sometimes we permit these pieces to overlap), which are allocated to nodes in a network. Nodes hold a dynamic state shared among their neighbors defined from the network structure. Natural partial or full row/column partitions and block partitions of the data
$\mathbf {A}, \mathbf {B},$
and
$\mathbf {C}$
are formulated by use of the vectorized matrix equation. We show that existing network flows for distributed linear algebraic equations can be extended to solve this special form of matrix equations over networks. A “consensus
$+$
projection
$+$
symmetrization” flow is also developed for equations with symmetry constraints on the matrix variables. We prove the convergence of these flows and obtain the fastest convergence rates that these flows can achieve regardless of the choices of node interaction strengths and network structures.