In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system.
is the coordinate transformation matrix defined as and whose submatrices are It can be observed that some of these matrices may have dimension zero.
By using results from controllability and observability, it can be shown that the transformed system
has matrices in the following form: This leads to the conclusion that A Kalman decomposition also exists for linear dynamical quantum systems.
Unlike classical dynamical systems, the coordinate transformation used in this variant requires to be in a specific class of transformations due to the physical laws of quantum mechanics.