Systemmatrix

Systemmatrix is a term used to denote a matrix that encodes the linear transformation describing a system’s internal dynamics or input-output behavior. In control theory, the system matrix typically refers to the A matrix in state-space representations. For a continuous-time linear system x' = A x + B u, y = C x + D u, A governs the evolution of the state x in the absence of input. The eigenvalues of A determine stability; the pair (A,B) may be analyzed for controllability, and (A,C) for observability. In discrete time, x_{k+1} = A x_k + B u_k, with similar interpretation.

Outside control, the term appears in imaging and signal processing, where the system matrix A maps an input signal or image x to observations y via y = A x. In tomography, A represents the data acquisition process; in numerical linear algebra, the system matrix is the coefficient matrix in Ax = b.

System matrices are typically large and sparse, and their properties such as dimension, sparsity pattern, conditioning, symmetry, and positive definiteness affect computational methods for simulation, estimation, and inversion. They are constructed from physical models, discretization schemes, and boundary conditions, and may be adapted during system identification to best fit recorded data.

See also: state-space representation, transfer function, eigenvalues, controllability, observability, Kalman filter, tomography, coefficient matrix.