Graph-structural Modeling: Schur Complement and Kron Reduction

The Kron reduction in the original presentation of Kron and the modern interpretation of the Kron reduction in the context of graph theory and applications are presented. An interpretation of Kron reduction is proposed in the context of more general and flexible graph structures of graph-structural modeling - meta-graphs. Since the Kron reduction uses the Schur complement of a block of a block matrix, a modern interpretation of the pseudo-Schur complement is presented, using the pseudo-inversion of the block in the case when the block is noninvertible. As an example, it is shown that the Kron reduction of the block of the Laplacian of the meta-arc of the meta-graph is an empty graph.