  In this paper, we study distributed methods for solving a Sylvester equation
in the form of AX+XB=C for matrices A, B, C$\in R^{n\times n}$ with X being the
unknown variable. The entries of A, B and C (called data) are partitioned into
a number of pieces (or sometimes we permit these pieces to overlap). Then a
network with a given structure is assigned, whose number of nodes is consistent
with the partition. Each node has access to the corresponding set of data and
holds a dynamic state. Nodes share their states among their neighbors defined
from the network structure, and we aim to design flows that can asymptotically
converge to a solution of this equation. The decentralized data partitions may
be resulted directly from networks consisting of physically isolated
subsystems, or indirectly from artificial and strategic design for processing
large data sets. Natural partial row/column partitions, full row/column
partitions and clustering block partitions of the data A, B and C are assisted
by the use of the vectorized matrix equation. We show that the existing
"consensus + projection" flow and the "local conservation + global consensus"
flow for distributed linear algebraic equations can be used to drive
distributed flows that solve this kind of equations. A "consensus + projection
+ symmetrization" flow is also developed for equations with symmetry
constraints on the solution matrices. We reveal some fundamental convergence
rate limitations for such flows regardless of the choices of node interaction
strengths and network structures. For a special case with B=A$^T$, where the
equation mentioned is reduced to a classical Lyapunov equation, we demonstrate
that by exploiting the symmetry of data, we can obtain flows with lower
complexity for certain partitions.
