Kopplungsmatrix

A Kopplungsmatrix, or coupling matrix, is a square matrix that encodes the strengths and structure of interactions between the components of a coupled system. It appears in mathematical models described by vectors of state variables, where the matrix multiplies the state or related quantities to produce terms that represent how one component influences another.

In linear models, the coupling matrix is part of the system matrix and captures mutual influence between subsystems. Depending on the formulation, it may refer to the full matrix that includes self-dynamics on the diagonal and interactions on the off-diagonal, or to the off-diagonal part that specifically models coupling.

Properties vary with context. It is typically square with a size equal to the number of subsystems or degrees of freedom; it may be real or complex, symmetric (for reciprocal coupling) or non-symmetric; and its sparsity reflects the locality of interactions.

Applications span engineering and physics. In mechanical vibrations, a Kopplungsmatrix describes how masses or modes influence each other in a network of springs. In electrical networks, it encodes mutual impedances or coupling between components. In multi-physics and control systems, it couples subsystems; in quantum mechanics, the matrix elements reflect transition amplitudes between states.

Analysis commonly uses eigenvalue decomposition. The eigenvalues and eigenvectors reveal the coupled normal modes and their frequencies, with larger off-diagonal terms indicating stronger interaction. In some cases the aim is to diagonalize the matrix to decouple the system or to identify dominant couplings.

Variants include time- or frequency-dependent coupling matrices and parametric forms. The concept is related to related matrices such as the stiffness, impedance, or adjacency matrices, but the term Kopplungsmatrix emphasizes interdependence between distinct components.