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