Diagrammatris

Diagrammatris, or diagram matrix, is a term used in several disciplines to refer to a matrix that encodes the relationships depicted by a diagram, such as a network, graph, or schematic. The idea is to translate a visual or conceptual diagram into a mathematical object that can be analyzed with linear algebra.

In graph theory, the most common diagrammatrices are the adjacency matrix, the incidence matrix, and the Laplacian matrix. An adjacency matrix records connections between nodes, with entries indicating the presence or weight of edges. An incidence matrix relates edges to their incident vertices, with rows representing edges and columns representing vertices. The Laplacian, defined as the degree matrix minus the adjacency matrix, is widely used in studying connectivity and diffusion on graphs.

Beyond graphs, diagrammatrices appear in other contexts where a diagram expresses relationships. For example, in circuit theory, a connectivity or incidence matrix can describe how components connect. In state machines or workflow diagrams, a transition matrix may encode allowable moves between states or stages. Weighted versions allow encoding strengths, costs, or capacities.

Construction and interpretation depend on the diagram type and the chosen representation. Edges may be directed or undirected, and multiple edges or self-loops may be allowed. The resulting matrix can be analyzed to deduce properties such as connectivity, shortest paths, flows, spectral characteristics, or steady states.

Terminology varies by language and field; the exact term diagrammatris is not universal in English, but the underlying concept—a matrix representation of a diagram—is standard across many disciplines. See also adjacency matrix, incidence matrix, Laplacian matrix, and graph theory.